Finally, we get that

$$ \begin{aligned} |\zeta(s-k,d)| &\ll_{r,r',\delta} r'^k + \frac{k!}{(2\pi)^k} (k+1)^n (1+r'^{2k}) + \frac{(2e)^k k!}{(2\pi)^k} (k+1)^{n+1} \\ &\ll_{r,r',\delta} \left(\frac{e}{\pi}\right)^k k!(k+1)^{r+1}(1+r'^{2k}). \end{aligned} $$

□

For all nonnegative integers $a \in \mathbb{N}_0$, we observe that the function $s \mapsto \frac{1}{\Gamma(s)(s+a)}$ is holomorphic on all $\mathbb{C}$. We get the following lemma:

**Lemma 2.3.4.** Let $r \in \mathbb{N}$. We have

$$ \begin{aligned} \forall s \in D_0(r), \quad & \left| \frac{1}{\Gamma(s)} \right| \ll 1 \\ \forall s \in D_0(r), \forall a \in \mathbb{N}_0, \quad & \left| \frac{1}{\Gamma(s)(s+a)} \right| \ll 1. \end{aligned} $$

*Proof.* As $\frac{1}{\Gamma}$ and $s \mapsto \frac{1}{\Gamma(s)s}$ are holomorphic on $\mathbb{C}$, we have uniformly on $D_0(r)$:

$$ \begin{aligned} \left| \frac{1}{\Gamma(s)} \right| &\ll 1 \\ \left| \frac{1}{\Gamma(s)s} \right| &\ll 1. \end{aligned} $$

Thus it is clear that

$$ \forall |s| > \frac{1}{2}, \quad \left| \frac{1}{s} \right| < 2. $$

One can then show the inequality required. Let $s \in \mathbb{C}$ such that $\sigma \in [-r, r]$.

If $|s+a| > \frac{1}{2}$, then

$$ \left| \frac{1}{\Gamma(s)(s+a)} \right| \ll 1. $$

If $|s+a| \le \frac{1}{2}$ then by the triangular inequality, we get $a \le |s| + 1/2 \le r + 1/2$. Using the functional equation of $\Gamma$ we get

$$ \frac{1}{\Gamma(s)(s+a)} = \frac{(s+a-1)...s}{\Gamma(s+a)(s+a)}, $$

thus we have the following inequality:

$$ \left| \frac{1}{\Gamma(s)(s+a)} \right| \ll (2r+1)...r. $$

As $a \le 1/2 + r$, we then get the following upper bound for the product in the numerator

$$ |(s+a-1)...s| \le (2r+1)! $$

Therefore, we find that

$$ \forall s \in \bar{D}_0(r), \quad \left| \frac{1}{\Gamma(s)(s+a)} \right| \ll 1. $$

□
Erdélyi's formula [EMOT81, §1.11] described in the following proposition is crucial for re-expressing integrands involving the Lerch zeta function that we defined in Definition 1.2.7:

**Proposition 2.3.5.** [Ono21, Lemma 2.2]

Let $u \in \mathbb{C}\setminus[-\infty, 0]$ such that $|u| < 2\pi$, $s \in \mathbb{C}\setminus\mathbb{N}$, and $d \in \mathbb{R}_+^*$. Then

$$e^{-du}\phi(e^{-u}, s, d) = \Gamma(1-s)u^{s-1} + \sum_{k=0}^{+\infty} \frac{(-u)^k}{k!} \zeta(s-k, d).$$

We aim to obtain an Erdélyi formula for $d \in H_0$. The bound we have on the term $\zeta(s-k, d)$ is not as good when $d$ is a complex number as when $d$ is real, thus we must assume an additional condition on the complex $u$:

**Proposition 2.3.6.** Let $d \in H_0$, $s \in \mathbb{C}\setminus\mathbb{N}$ and $u \in \mathbb{C}\setminus[-\infty, 0]$ such that $|u| \ll 1$. then

$$e^{-du}\phi(e^{-u}, s, d) = \Gamma(1-s)u^{s-1} + \sum_{k=0}^{+\infty} \frac{(-u)^k}{k!} \zeta(s-k, d) \quad (2.6)$$

*Proof.* Let $r' \in [1, +\infty[$ be a real number, we will aim to prove the result for $d \in H_{1/r'} \cap D_0(r')$. Let $u \in \mathbb{C}\setminus[-\infty, 0]$ such that $|u| < \frac{\pi}{e(1+r')^2}$. We fix $s \in \mathbb{C}\setminus\mathbb{N}$, and set $r := \lceil|s|\rceil$. We already know that the formula we wish to prove holds when $d \in \mathbb{R}_+^*$. We then aim to prove this formula for all $d \in H_0$ through analytic continuation. To do this, we first need to show that the different terms in the expression of the desired formula are holomorphic in the variable $d$ over $H_0$.

We know that the map $d \in H_0 \mapsto e^{-du}\phi(e^{-u}, s, d)$ is holomorphic [EMOT81, §1.11]; it remains to show that the series appearing in the formula is holomorphic as well. We already know that each terms $d \in H_0 \mapsto \zeta(s-k, d)$ are holomorphic over $H_0$ for every integer $k \in \mathbb{N}_0$. To prove this, it is enough to refer to the formula obtained by Maclaurin's expansion of the Hurwitz zeta function, found in the proof of Proposition 1.2.6. What remains is to show that the series appearing in (2.6) converges normally in $d$ over $\bar{H}_{1/r'} \cap D_0(r')$. We have the following bound

$$\forall k \in \mathbb{N}_0, \quad |\zeta(s-k, d)| \ll_{r, r', \delta} \left(\frac{e}{\pi}\right)^k k!(k+1)^{r+1}(1+r')^{2k},$$

thus, we find that the general term of the series 2.6 has the following upper bound

$$\frac{|\zeta(s-k, d)|}{k!} |u|^k \ll_{r, r', \delta} \left(u \frac{e}{\pi} (1+r')^2\right)^k (k+1)^{r+1},$$

but the power series $\sum_{k=0}^{+\infty} \left(\frac{e}{\pi}(1+r')^2\right)^k k^{r+1}z^k$ is absolutely convergent in the open disc $\{z \in \mathbb{C}, |z| < \frac{\pi}{e(1+r')^2}\}$.

We then find that the series $\sum_{k=0}^{+\infty} \frac{\zeta(s-k, d)}{k!} (-u)^k$ is normally convergent in $d$ over $H_{1/r'} \cap D_0(r')$.

Therefore, the formula (2.6) holds for all $d \in H_0$ by analytic continuation, via Proposition 2.3.5. $\square$

## 2.3.2 Preliminary Results

In this section, we wish to establish a Crandall expansion for the multizeta function.

$$Z(\mathbf{s}, \mathbf{s}') := \sum_{\mathbf{n} \in \mathbb{N}_0^P} \prod_{p=1}^P (n_p + d_p)^{-s_p} \prod_{q=1}^Q (l_q(\mathbf{n}) + d'_q)^{-s'_q}.$$
We recall Definition 1.3.1: Let $\nu \in H_0$, $\theta > 0$, $s \in \mathbb{C}$. We defined

$$ 
\begin{aligned}
\Gamma(s, \theta, \nu) &= \int_{\theta}^{+\infty} e^{-\nu y} y^{s-1} dy, \\
\gamma(s, \theta, \nu) &= \int_{0}^{\theta} e^{-\nu y} y^{s-1} dy, \quad \text{when } \mathrm{Re}(s) > 0.
\end{aligned}
$$

Let's note that the integral defining $\Gamma(s, \theta, \nu)$ is absolutely convergent for all $s \in \mathbb{C}$, but the one defining $\gamma(s, \theta, \nu)$ is absolutely convergent only for $s \in H_0$.

**Proposition 2.3.7.** Let $\theta > 0$. Then the functions $(s, \nu) \in \mathbb{C} \times H_0 \mapsto \Gamma(s, \theta, \nu)$ and $(s, \nu) \in H_0 \times H_0 \mapsto \gamma(s, \theta, \nu)$ are holomorphic respectively on $\mathbb{C} \times H_0$ and on $H_0 \times H_0$.

*Proof.* The proposition follows from the theorem of holomorphy under the integral sign. $\square$

**Proposition 2.3.8.** Let $\nu \in H_0$, $\theta > 0$. We have

$$ \forall s \in H_0, \quad \Gamma(s, \theta, \nu) + \gamma(s, \theta, \nu) = \nu^{-s} \Gamma(s). $$

*Proof.* Let's assume that $\nu \in \mathbb{R}_+^*$. By a change of variables we get:

$$ 
\begin{aligned}
\Gamma(s, \theta, \nu) &= \nu^{-s} \int_{\nu\theta}^{+\infty} e^{-x} x^{s-1} dx, \\
\gamma(s, \theta, \nu) &= \nu^{-s} \int_{0}^{\nu\theta} e^{-x} x^{s-1} dx, \quad \text{when } \mathrm{Re}(s) > 0.
\end{aligned}
$$

Summing up these 2 terms, we have

$$ \Gamma(s, \theta, \nu) + \gamma(s, \theta, \nu) = \nu^{-s} \Gamma(s). $$

Let $s \in H_0$. The function $\nu \in H_0 \mapsto \Gamma(s, \theta, \nu) + \gamma(s, \theta, \nu) - \nu^{-s}\Gamma(s)$ is holomorphic, and vanish in $s \in \mathbb{R}_+^*$. By the isolated zero theorem, we get the final result. $\square$

**Corollary 2.3.9.** The function $s \mapsto \gamma(s, \theta, \nu)$ is meromorphic on all $\mathbb{C}$, having poles (which are simple) only at nonpositive integers, and the residue at $s = -n$ is $\frac{(-1)^n \nu^n}{n!}$.

**Lemma 2.3.10.** Let $\theta > 0$ be a positive number, and $\nu \in H_0$. Let $s \in \mathbb{C}$. We have the following bound:

$$ |\Gamma(s, \theta, \nu)| \le 2^{\sigma} e^{-\mathrm{Re}(\nu)\theta/2} \Gamma\left(\sigma, \frac{\theta}{2}, \mathrm{Re}(\nu)\right). $$

*Proof.* By the change of variables $x = \frac{x'}{2}$, we found

$$ \Gamma(s, \theta, \nu) = 2^s \int_{\theta/2}^{+\infty} e^{-2\nu x} x^{s-1} dx. $$

As $|e^{-\nu x}| = e^{-\mathrm{Re}(\nu)x}$ for all $x \in \mathbb{R}_+$ and for all complex numbers $\nu \in H_0$ we find the following bound:

$$ 
\begin{aligned}
|\Gamma(s, \theta, \nu)| &\le 2^s \int_{\theta/2}^{+\infty} e^{-\mathrm{Re}(\nu)x} e^{-\mathrm{Re}(\nu)x} x^{\sigma-1} dx \\
&\le 2^{\sigma} \int_{\theta/2}^{+\infty} e^{-\mathrm{Re}(\nu)\theta/2} e^{-\mathrm{Re}(\nu)x} x^{\sigma-1} dx,
\end{aligned}
$$

which concludes the proof. $\square$
We now study the product $\prod_{q=1}^{Q} \Gamma(s'_q)Z(s, s')$. Let $q \in [1, Q]$, $\theta > 0$, and $s'_q \in H_0$. By Proposition 2.3.8, we get

$$ \Gamma(s'_q)(l_q(\mathbf{n}) + d'_q)^{-s'_q} = \Gamma(s'_q, \theta, (l_q(n) + d'_q)) + \gamma(s'_q, \theta, (l_q(\mathbf{n}) + d'_q)). $$

We fix a $P$-tuple $\mathbf{n} \in \mathbb{N}_0^P$, and complex numbers $(\mathbf{s}, \mathbf{s}') \in \mathbb{C}^P \times H_0^Q$. We thus obtain the following equality by distributing the sum above over the subsequent product:

$$ \begin{aligned} \prod_{q=1}^{Q} \Gamma(s'_q)(l_q(\mathbf{n}) + d'_q)^{-s'_q} &= \prod_{q=1}^{Q} \left[ \Gamma(s'_q, \theta, l_q(\mathbf{n}) + d'_q) + \gamma(s'_q, \theta, l_q(\mathbf{n}) + d'_q) \right] \\ &= \sum_{\emptyset \neq \mathcal{Q} \subset [1, Q]} \prod_{q \in \mathcal{Q}} \Gamma(s'_q, \theta, l_q(\mathbf{n}) + d'_q) \prod_{q \in \mathcal{Q}^c} \gamma(s'_q, \theta, l_q(\mathbf{n}) + d'_q) \\ &\quad + \prod_{q=1}^{Q} \gamma(s'_q, \theta, l_q(\mathbf{n}) + d'_q) \\ &= \sum_{\emptyset \neq \mathcal{Q} \subset [1, Q]} \prod_{q \in \mathcal{Q}} \Gamma(s'_q, \theta, l_q(\mathbf{n}) + d'_q) \prod_{q \in \mathcal{Q}^c} \left( (l_q(\mathbf{n}) + d'_q)^{-s'_q} \Gamma(s'_q) - \Gamma(s'_q, \theta, l'_q(\mathbf{n}) + d'_q) \right) \\ &\quad + \prod_{q=1}^{Q} \gamma(s'_q, \theta, l_q(\mathbf{n}) + d'_q). \end{aligned} $$

By distributing the terms $(l_q(\mathbf{n}) + d'_q)^{-s'_q} \Gamma(s'_q) - \Gamma(s'_q, \theta, l'_q(\mathbf{n}) + d'_q)$, we get

$$ \begin{aligned} \prod_{q=1}^{Q} \Gamma(s'_q)(l_q(\mathbf{n}) + d'_q)^{-s'_q} &= \sum_{\emptyset \neq \mathcal{Q} \subset [1, Q]} \sum_{A_{\mathcal{Q}} \subset \mathcal{Q}^c} \prod_{q \in \mathcal{Q}} \Gamma(s'_q, \theta, l_q(\mathbf{n}) + d'_q) \prod_{q \in A_{\mathcal{Q}}} (l_q(\mathbf{n}) + d'_q)^{-s'_q} \\ &\quad \prod_{q \in A_{\mathcal{Q}}} \Gamma(s'_q) \prod_{q \in \mathcal{Q}^c \setminus A_{\mathcal{Q}}} (-1) \Gamma(s'_q, \theta, l_q(\mathbf{n}) + d'_q) \\ &\quad + \prod_{q=1}^{Q} \gamma(s'_q, \theta, l_q(\mathbf{n}) + d'_q). \end{aligned} $$

Let $(x_p)_{p \in [1, P]} \in H_0^P$, then by the previous equality we have that

$$ \begin{aligned} \prod_{q=1}^{Q} \Gamma(s'_q)(l_q(\mathbf{n}) + d'_q)^{-s'_q} \prod_{p=1}^{P} x_p^{-s_p} = & \\ & \sum_{\emptyset \neq \mathcal{Q} \subset [1, Q]} \sum_{A_{\mathcal{Q}} \subset \mathcal{Q}^c} (-1)^{|\mathcal{Q}^c| - |A_{\mathcal{Q}}|} \prod_{p=1}^{P} x_p^{-s_p} \prod_{q \in \mathcal{Q} \cup (\mathcal{Q}^c \setminus A_{\mathcal{Q}})} \Gamma(s'_q, \theta, l_q(\mathbf{n}) + d'_q) \left( \prod_{q \in A_{\mathcal{Q}}} (l_q(\mathbf{n}) + d'_q)^{-s'_q} \Gamma(s'_q) \right) \\ & + \prod_{p=1}^{P} x_p^{-s_p} \prod_{q=1}^{Q} \gamma(s'_q, \theta, l_q(\mathbf{n}) + d'_q). \end{aligned} $$

As $\mathcal{Q} \cup (\mathcal{Q}^c \setminus A_{\mathcal{Q}}) = [1, Q] \setminus A_{\mathcal{Q}}$, by replacing $x_p$ by $n_p + d_p$ we get:

$$ \begin{aligned} \prod_{q=1}^{Q} \Gamma(s'_q)(l_q(\mathbf{n}) + d'_q)^{-s'_q} \prod_{p=1}^{P} (n_p + d_p)^{-s_p} = & \quad (2.7) \\ & \sum_{\emptyset \neq \mathcal{Q} \subset [1, Q]} \sum_{A_{\mathcal{Q}} \subset \mathcal{Q}^c} (-1)^{|\mathcal{Q}^c| - |A_{\mathcal{Q}}|} \prod_{p=1}^{P} (n_p + d_p)^{-s_p} \prod_{q \in [1, Q] \setminus A_{\mathcal{Q}}} \Gamma(s'_q, \theta, l_q(\mathbf{n}) + d'_q) \left( \prod_{q \in A_{\mathcal{Q}}} (l_q(\mathbf{n}) + d'_q)^{-s'_q} \Gamma(s'_q) \right) \\ & + \prod_{p=1}^{P} (n_p + d_p)^{-s_p} \prod_{q=1}^{Q} \gamma(s'_q, \theta, l_q(\mathbf{n}) + d'_q). \end{aligned} $$
Summing the last equality over $n_p \in \mathbb{N}_0$ for all $p \in [1, P]$, we shall prove later on that for $(\mathbf{s}, \mathbf{s}') \in \mathcal{D}$ (with $\mathcal{D}$ the set defined in Proposition 2.1.1), we get

$$ Z(\mathbf{s}, \mathbf{s}') = \sum_{\emptyset \neq \mathcal{Q} \subset [1, Q]} \sum_{A_{\mathcal{Q}} \subset \mathcal{Q}^c} (-1)^{|\mathcal{Q}^c| - |A_{\mathcal{Q}}|} \prod_{q \in [1, Q] \setminus A_{\mathcal{Q}}} \frac{1}{\Gamma(s'_q)} \left( \sum_{\mathbf{n} \in \mathbb{N}_0^P} \prod_{p=1}^P (n_p + d_p)^{-s_p} \right. \\ \left. \cdot \prod_{q \in A_{\mathcal{Q}}} (l_q(\mathbf{n}) + d'_q)^{-s'_q} \prod_{q \in [1, Q] \setminus A_{\mathcal{Q}}} \Gamma(s'_q, \theta, l_q(\mathbf{n}) + d'_q) \right) \\ + \frac{1}{\prod_{q=1}^Q \Gamma(s'_q)} \sum_{\mathbf{n} \in \mathbb{N}_0^P} \prod_{p=1}^P (n_p + d_p)^{-s_p} \prod_{q=1}^Q \gamma(s'_q, \theta, l_q(\mathbf{n}) + d'_q). $$

By formula 2.7, it is enough to show that the previous series converge absolutely. We shall study those series in the two following propositions:

**Proposition 2.3.11.** Let $\theta > 0$, and $(\mathbf{s}, \mathbf{s}') \in \mathbb{C}^P \times \mathbb{C}^Q$. We set

$$ K(\theta, \mathbf{s}, \mathbf{s}') = \sum_{\emptyset \neq \mathcal{Q} \subset [1, Q]} \sum_{A_{\mathcal{Q}} \subset \mathcal{Q}^c} (-1)^{|\mathcal{Q}^c| - |A_{\mathcal{Q}}|} \prod_{q \in [1, Q] \setminus A_{\mathcal{Q}}} \frac{1}{\Gamma(s'_q)} \left( \sum_{\mathbf{n} \in \mathbb{N}_0^P} \prod_{p=1}^P (n_p + d_p)^{-s_p} \right. \\ \left. \cdot \prod_{q \in A_{\mathcal{Q}}} (l_q(\mathbf{n}) + d'_q)^{-s'_q} \prod_{q \in [1, Q] \setminus A_{\mathcal{Q}}} \Gamma(s'_q, \theta, l_q(\mathbf{n}) + d'_q) \right). \quad (2.8) $$

Then the series above converges normally on all compact sets of $\mathbb{C}^P \times \mathbb{C}^Q$. Moreover, the function $(\mathbf{s}, \mathbf{s}') \mapsto K(\theta, \mathbf{s}, \mathbf{s}')$ is holomorphic on $\mathbb{C}^P \times \mathbb{C}^Q$, and $K(\theta, -\mathbf{N}, -\mathbf{N}') = 0$ for all $Q+T$-tuple $(\mathbf{N}, \mathbf{N}') \in \mathbb{N}_0^P \times \mathbb{N}_0^Q$.

*Proof.* Let $\theta > 0$ be a real number. In order to prove that $K$ is holomorphic in the variables $(\mathbf{s}, \mathbf{s}')$ on all $\mathbb{C}^P \times \mathbb{C}^Q$, it is enough to show that the series defining $K(\theta, \mathbf{s}, \mathbf{s}')$ converges normally on all compact sets. The function $K(\theta, \mathbf{s}, \mathbf{s}')$ is a finite sum of the following terms

$$ (-1)^{|\mathcal{Q}^c| - |A_{\mathcal{Q}}|} \prod_{q \in [1, Q] \setminus A_{\mathcal{Q}}} \frac{1}{\Gamma(s'_q)} \left( \sum_{\mathbf{n} \in \mathbb{N}_0^P} \prod_{p=1}^P (n_p + d_p)^{-s_p} \prod_{q \in A_{\mathcal{Q}}} (l_q(\mathbf{n}) + d'_q)^{-s'_q} \prod_{q \in [1, Q] \setminus A_{\mathcal{Q}}} \Gamma(s'_q, \theta, l_q(\mathbf{n}) + d'_q) \right), \quad (2.9) $$

with $\emptyset \neq \mathcal{Q} \subset [1, Q]$, and $A_{\mathcal{Q}} \subset \mathcal{Q}^c$. We see that the functions $\frac{1}{\Gamma(s'_q)}$, and $\Gamma(s'_q, \theta, l_q(\mathbf{n}) + d'_q)$ are well defined, and holomorphic on $\mathbb{C}$. We define

$$ T_{\mathcal{Q}, A_{\mathcal{Q}}}(\theta, \mathbf{n}, \mathbf{s}, \mathbf{s}') = \prod_{p=1}^{P} (n_p + d_p)^{-s_p} \prod_{q \in A_{\mathcal{Q}}} (l_q(\mathbf{n}) + d'_q)^{-s'_q} \prod_{q \in [1, Q] \setminus A_{\mathcal{Q}}} \Gamma(s'_q, \theta, l_q(\mathbf{n}) + d'_q), \quad (\mathbf{n} \in \mathbb{N}_0^P) \quad (2.10) $$

as the general term of the series in (2.9).

By Lemma 2.3.10, we get that, for all $q \in [1, Q]$ and all $\mathbf{n} \in \mathbb{N}_0^P$,

$$ |\Gamma(s'_q, \theta, l_q(\mathbf{n}) + d'_q)| \le 2^{\sigma'_q} e^{-(\mathrm{Re}(l_q(\mathbf{n})+d'_q))^{\theta/2}} \Gamma(\sigma'_q, \theta/2, \mathrm{Re}(l_q(\mathbf{n}) + d'_q)). $$

By the previous bound, and using the fact that the following function $x \in \mathbb{R}_+^* \mapsto \Gamma(\sigma, \theta, x)$ is decreasing, for $\theta > 0$, we get

$$ |\Gamma(s'_q, \theta, l_q(\mathbf{n}) + d'_q)| \le 2^{\sigma'_q} e^{-(\mathrm{Re}(l_q(\mathbf{n})+d'_q))^{\theta/2}} \Gamma(\sigma'_q, \theta/2, \mathrm{Re}(d'_q)). $$
Finally, for all complex numbers $z \in H_0$, $s = \sigma + i\tau \in \mathbb{C}$, we get $|z^s| = |z|^\sigma e^{-\arg(z)\tau}$. As $z \in H_0$, we then find that $\arg(z) \in [-\pi/2, \pi/2]$, and thus

$$|z^s| \le |z|^\sigma e^{\tau|\pi/2}}.$$

From all the above, we get the following bound for the general term:

$$ \begin{aligned} |T_{Q,A_Q}(\theta, \mathbf{n}, \mathbf{s}, \mathbf{s}')| & \le \left| \prod_{p=1}^{P} (n_p + d_p)^{-s_p} \prod_{q \in A_Q} (l_q(\mathbf{n}) + d'_q)^{-s_q} \right| \prod_{q \in [1, Q] \setminus A_Q} 2^{\sigma'_q} e^{-(\mathrm{Re}(l_q(\mathbf{n})+d'_q))^{\theta/2}} \Gamma(\sigma'_q, \theta/2, \mathrm{Re}(d'_q)) \\ & \le \prod_{q \in [1, Q] \setminus A_Q} e^{-(\mathrm{Re}(l_q(\mathbf{n})+d'_q))\theta/2} 2^{\sigma'_q} \Gamma(\sigma'_q, \theta/2, d'_q) \\ & \qquad \cdot \prod_{p=1}^{P} e^{\tau_p|\pi/2}} |n_p + d_p|^{-\sigma_p} \prod_{q \in A_Q} e^{\tau'_q|\pi/2}} |(l_q(\mathbf{n}) + d'_q|^{-\sigma'_q}. \end{aligned} \quad (2.11) $$

Let's note that, if all the coefficients $c_{p,q}$, $d_p$ and $d'_q$ are real numbers, then we can get rid of the $e^{\tau_p|\pi/2}}$ and $e^{\tau'_q|\pi/2}}$ terms in the last inequality.

Because the set $\mathcal{Q}$ is non empty, and that $A_Q \subset \mathcal{Q}^c$, then this last set contains at least an integer $j \in A_Q$, and by our assumptions, $l_j(\mathbf{n}) = \sum_{p=1}^P c_{j,p} n_p$, with $c_{j,p} \in H_0$ for all $p \in [1, P]$. We also have that $d'_q \in H_0$ for all $q \in [1, Q]$. Therefore,

$$\prod_{q \in \mathcal{Q}} e^{-\mathrm{Re}(l_q(\mathbf{n})+d'_q)\theta/2} \le e^{-\mathrm{Re}(l_j(\mathbf{n}))\theta/2}.$$

Let $R > 0$, and $(\mathbf{s}, \mathbf{s}') \in D_0(R)^{P+Q}$. We then found that $(\boldsymbol{\sigma}, \boldsymbol{\sigma}') \in [-R, R]^{P+Q}$ and $(\boldsymbol{\tau}, \boldsymbol{\tau}') \in [-R, R]^{P+Q}$. The map $x \in [-R, R] \mapsto \Gamma(x, \theta, \nu)$ being a continuous function for all $\theta > 0$, and all $\nu \in H_0$, we found the following bound, uniform in $(\mathbf{s}, \mathbf{s}') \in D_0(R)^P \times D_0(R)^Q$,

$$\prod_{q \in [1, Q] \setminus A_Q} \Gamma(\sigma'_q, \theta/2, \mathrm{Re}(d'_q)) \ll 1.$$

By the bound 2.11, we found that for all $(\mathbf{s}, \mathbf{s}') \in D_0(R)^P \times D_0(R)^Q$,

$$|T_{Q,A_Q}(\theta, \mathbf{n}, \mathbf{s}, \mathbf{s}')| \ll 2^{(Q-|A_Q|)R} \prod_{p=1}^{P} e^{\pi/2R} e^{-\mathrm{Re}(c_{j,p})n_p\theta/2} |n_p + d_p|^{-\sigma_p} \cdot \prod_{q \in A_Q} e^{\pi/2R} |l_q(\mathbf{n}) + d'_q|^{-\sigma'_q}. \quad (2.12)$$

As $\theta > 0$, and $\mathrm{Re}(c_{j,p}) > 0$ for all $p \in [1, P]$, we know that the series

$$\sum_{\mathbf{n} \in \mathbb{N}_0^P} \prod_{p=1}^P e^{-\mathrm{Re}(c_{j,p})n_p\theta/2} |n_p + d_p|^{-\sigma_p} \prod_{q \in A_Q} |l_q(\mathbf{n}) + d'_q|^{-\sigma'_q}$$

converges normally on all compact sets of $\mathbb{C}^P \times \mathbb{C}^Q$. Thus, we found that the series $\sum_{\mathbf{n} \in \mathbb{N}_0^P} T_{Q,A_Q}(\theta, \mathbf{n}, \mathbf{s}, \mathbf{s}')$ converges normally on $D_0(R)^{P+Q}$. We then get the fact that $K$ is holomorphic on $\mathbb{C}^P \times \mathbb{C}^Q$.

Finally, we see that, for all $\emptyset \ne \mathcal{Q} \subset [1, Q]$ and for all $A_Q \subset \mathcal{Q}^c$, we have that $[1, Q] \setminus A_Q \ne \emptyset$. Therefore, the product $\prod_{q \in [1, Q] \setminus A_Q} \frac{1}{\Gamma(s'_q)}$ is non-empty. In particular, when we take the values of $K(\theta, \mathbf{s}, \mathbf{s}')$ at $\mathbf{s}' = -\mathbf{N}'$, we get 0. $\square$

**Remark 2.3.12.** By evaluating formula 2.8 and replacing $\mathbf{s}$ by $(-N_p + \mu_p s)_{p \in [1, P]}$, and $\mathbf{s}'$ by $(-N'_q + \mu'_q s)_{q \in [1, Q]}$, we recover expression 2.4 in Proposition 2.2.4.
**Proposition 2.3.13.** Let $\theta > 0$, and $(\mathbf{s}, \mathbf{s}') \in \mathcal{D} \cap (\mathbb{C}^P \times H_0^Q)$, with $\mathcal{D}$ the set defined in Proposition 2.1.1. We define

$$J(\theta, \mathbf{s}, \mathbf{s}') = \frac{1}{\prod_{q=1}^{Q} \Gamma(s'_q)} \sum_{\mathbf{n} \in \mathbb{N}_0^P} \prod_{p=1}^{P} (n_p + d_p)^{-s_p} \prod_{q=1}^{Q} \gamma(s'_q, \theta, l_q(\mathbf{n}) + d'_q) \quad (2.13)$$

The preceding series is normally convergent on any compact subset of $\mathcal{D} \cap (\mathbb{C}^P \times H_0^Q)$. In particular, the function $(\mathbf{s}, \mathbf{s}') \mapsto J(\theta, \mathbf{s}, \mathbf{s}')$ is holomorphic on $\mathcal{D} \cap (\mathbb{C}^P \times H_0^Q)$.

*Proof.* Let $\theta > 0$, and $R > 0$ be positive real numbers. We use the notations from the previous proposition, particularly formula 2.10 to describe the general term of the series present in $K(\theta, \mathbf{s}, \mathbf{s}')$. By equality 2.7, we know that, for all $\mathbf{n} \in \mathbb{N}_0^P$ and $(\mathbf{s}, \mathbf{s}') \in \mathbb{C}^P \times H_1^Q$:

$$\frac{1}{\prod_{q=1}^{Q} \Gamma(s'_q)} \prod_{p=1}^{P} (n_p + d_p)^{-s_p} \prod_{q=1}^{Q} \gamma(s'_q, \theta, l_q(\mathbf{n}) + d'_q) = - \left( \sum_{\emptyset \neq \mathcal{Q} \subset [1, Q]} \sum_{A_{\mathcal{Q}} \subset \mathcal{Q}^c} \frac{(-1)^{|\mathcal{Q}^c| - |A_{\mathcal{Q}}|}}{\prod_{q \in [1, Q] \setminus A_{\mathcal{Q}}} \Gamma(s'_q)} T_{\mathcal{Q}, A_{\mathcal{Q}}} (\theta, \mathbf{n}, \mathbf{s}, \mathbf{s}') \right) + \prod_{p=1}^{P} (n_p + d_p)^{-s_p} \prod_{q=1}^{Q} (l_q(\mathbf{n}) + d'_q)^{-s'_q}$$

In the proof of Proposition 2.3.11, we showed using inequality 2.12 that the series with general term $T_{\mathcal{Q}, A_{\mathcal{Q}}}(\theta, \mathbf{n}, \mathbf{s}, \mathbf{s}')$ is normally convergent on any compact subset of $\mathbb{C}^P \times \mathbb{C}^Q$. We proved in Proposition 2.1.1 that the series of functions in $(\mathbf{s}, \mathbf{s}')$

$$\sum_{\mathbf{n} \in \mathbb{N}_0^P} \prod_{p=1}^{P} (n_p + d_p)^{-s_p} \prod_{q=1}^{Q} (l_q(\mathbf{n}) + d'_q)^{-s'_q}$$

is normally convergent on any compact subset of $\mathcal{D}$. Thus, we obtain that the series

$$\sum_{\mathbf{n} \in \mathbb{N}_0^P} \frac{1}{\prod_{q=1}^{Q} \Gamma(s'_q)} \prod_{p=1}^{P} (n_p + d_p)^{-s_p} \prod_{q=1}^{Q} \gamma(s'_q, \theta, l_q(\mathbf{n}) + d'_q)$$

is normally convergent on any compact subset of $\mathcal{D} \cap (\mathbb{C}^P \times H_0^Q)$. $\square$

**Proposition 2.3.14.** Let $\theta > 0$, and $(\mathbf{s}, \mathbf{s}') \in \mathcal{D} \cap (\mathbb{C}^P \times H_0^Q)$. We have:

$$Z(\mathbf{s}, \mathbf{s}') = K(\theta, \mathbf{s}, \mathbf{s}') + J(\theta, \mathbf{s}, \mathbf{s}'), \quad (2.14)$$

Furthermore, the function $(\mathbf{s}, \mathbf{s}') \mapsto J(\theta, \mathbf{s}, \mathbf{s}')$ admits a meromorphic extension on $\mathbb{C}^P \times \mathbb{C}^Q$.

*Proof.* The first equality follows from the last two propositions. Since $(\mathbf{s}, \mathbf{s}') \mapsto Z(\mathbf{s}, \mathbf{s}')$ is meromorphic on $\mathbb{C}^P \times \mathbb{C}^Q$, and $(\mathbf{s}, \mathbf{s}') \mapsto K(\theta, \mathbf{s}, \mathbf{s}')$ is holomorphic on the same space, it follows that $(\mathbf{s}, \mathbf{s}') \mapsto J(\theta, \mathbf{s}, \mathbf{s}')$ is meromorphic on $\mathbb{C}^P \times \mathbb{C}^Q$. $\square$

**Corollary 2.3.15.** The meromorphic function $(\mathbf{s}, \mathbf{s}') \mapsto Z(\mathbf{s}, \mathbf{s}')$ and the meromorphic function $(\mathbf{s}, \mathbf{s}') \mapsto J(\theta, \mathbf{s}, \mathbf{s}')$ share exactly the same singularities, and the multiplicity of their polar divisor is identical at the same singularity. Moreover, $(\mathbf{s}, \mathbf{s}') \mapsto J(\theta, \mathbf{s}, \mathbf{s}')$ is holomorphic on the domain $\mathcal{D}$, where $\mathcal{D}$ is the set defined in Proposition 2.1.1.

To compute the values of $Z$ at nonpositive integers, it remains to study $J(\theta, \mathbf{s}, \mathbf{s}')$. To do this, we will express this term as an integral of a Lerch zeta function, and then simplify the integrand using an Erdélyi formula for each term of the product.

Let's recall the following notations for the functions $l_p^*$: Let $p \in [1, P]$. We consider $l_p^* : \mathbb{R}^Q \to \mathbb{C}$ the following linear form

$$l_p^*(x) := \sum_{q=1}^{Q} c_{q,p} x_q.$$
By taking the previous expression of $J(\theta, s, s')$ for $(s, s') \in \mathcal{D} \cap (\mathbb{C}^P \times H_0^Q)$, we get that

$$
\begin{align}
J(\theta, \mathbf{s}, \mathbf{s}') &= \frac{1}{\prod_{q=1}^{Q} \Gamma(s'_q)} \sum_{\mathbf{n} \in \mathbb{N}_0^P} \prod_{p=1}^{P} (n_p + d_p)^{-s_p} \int_0^\theta \dots \int_0^\theta \prod_{q=1}^{Q} e^{-(l_q(\mathbf{n})+d'_q)x_q} x_q^{s'_q-1} dx_q \tag{2.15} \\
&= \frac{1}{\prod_{q=1}^{Q} \Gamma(s'_q)} \sum_{\mathbf{n} \in \mathbb{N}_0^P} \prod_{p=1}^{P} (n_p + d_p)^{-s_p} \int_0^\theta \dots \int_0^\theta \prod_{p=1}^{P} e^{-l_p^*(x)n_p} \prod_{q=1}^{Q} e^{-d'_q x_q} x_q^{s'_q-1} dx_q.
\end{align}
$$

**Definition 2.3.16.** Let $(\mathbf{N}, \mathbf{N}') \in \mathbb{N}_0^P \times \mathbb{N}_0^Q$, and $(\boldsymbol{\mu}, \boldsymbol{\mu}') \in \mathbb{R}_+^P \times \mathbb{R}_+^Q$. We define for $s \in \mathbb{C}$,

$$
J_{\mathbf{N}, \mathbf{N}'}(\theta, s) := J(\theta, -\mathbf{N} + \boldsymbol{\mu}s, -\mathbf{N}' + \boldsymbol{\mu}'s)
$$

$$
K_{\mathbf{N}, \mathbf{N}'}(\theta, s) := K(\theta, -\mathbf{N} + \boldsymbol{\mu}s, -\mathbf{N}' + \boldsymbol{\mu}'s).
$$

By (2.14), for $s \in \mathbb{C}$ such that $(-\mathbf{N} + \boldsymbol{\mu}s, -\mathbf{N}' + \boldsymbol{\mu}'s) \in \mathcal{D} \cap (\mathbb{C}^P \times H_0^Q)$, we have

$$
Z_{\mathbf{N}, \mathbf{N}'}(s) = J_{\mathbf{N}, \mathbf{N}'}(\theta, s) + K_{\mathbf{N}, \mathbf{N}'}(\theta, s).
$$

By definition of the convergence domain $\mathcal{D}$ in Proposition 1.3.9, we get that the previous assumption on $s$ is verified if

$$
\begin{align*}
\forall \mathcal{P} \subset [1, P], \quad & -|\mathbf{N}|_{\mathcal{P}} - |\mathbf{N}'| + (|\boldsymbol{\mu}|_{\mathcal{P}} + |\boldsymbol{\mu}'|)\sigma > |\mathcal{P}| \\
\forall q \in [1, Q], \quad & -N'_q + \mu'_q \sigma > 0.
\end{align*}
$$

We define $\sigma_0(\mathbf{N}, \mathbf{N}') := \max \left( \frac{N'_1}{\mu'_1}, \dots, \frac{N'_Q}{\mu'_Q}, \max_{\mathcal{P} \subset [1, P]} \left( \frac{1+|\mathcal{P}|+|\mathbf{N}'|+|\mathbf{N}|_{\mathcal{P}}}{|\boldsymbol{\mu}'|+|\boldsymbol{\mu}|_{\mathcal{P}}} \right) \right)$. We then get the following proposition:

**Proposition 2.3.17.** For all $s \in H_{\sigma_0(\mathbf{N},\mathbf{N}')}$, we have

$$
Z_{\mathbf{N}, \mathbf{N}'}(s) = J_{\mathbf{N}, \mathbf{N}'}(\theta, s) + K_{\mathbf{N}, \mathbf{N}'}(\theta, s).
$$

**Remark 2.3.18.** 1) Rather than using the notation $s = \sigma + i\tau \in H_{\sigma_0(\mathbf{N},\mathbf{N}')}$, we prefer to denote $\sigma \gg_{\mathbf{N},\mathbf{N}'} 1$, or even sometimes $\sigma \gg 1$.
$\mu, \mu'$

2) Here we use the fact that $\mu'_q > 0$ for all $1 \le q \le Q$. One should be able to generalize this approach by considering directions $\mu'_q \in H_0$, however, this would complicate the reasoning for this chapter.

We already know from Proposition 2.3.11 that $s \mapsto K_{\mathbf{N},\mathbf{N}'}(\theta, s)$ is holomorphic and vanishes at $s=0$. What remains to be studied is the term $J_{\mathbf{N},\mathbf{N}'}(\theta, s)$. The key point in studying this function is Erdélyi's formula.

**Proposition 2.3.19.** Let $(\mathbf{N}, \mathbf{N}') \in \mathbb{N}_0^P \times \mathbb{N}_0^Q$, and $(\boldsymbol{\mu}, \boldsymbol{\mu}') \in \mathbb{R}_+^P \times \mathbb{R}_+^Q$ Let $s \in \mathbb{C}$ such that $\mu_p s \notin \mathbb{N}$ for all $p \in [1, P]$, and $\sigma \gg 1$. We then have

$$
J_{\mathbf{N}, \mathbf{N}'}(\theta, s) = \frac{1}{\prod_{q=1}^{Q} \Gamma(-N'_q + \mu'_q s)} \int_0^\theta \dots \int_0^\theta \prod_{p=1}^{P} e^{-d_p l_p^*(x)} \phi(e^{-l_p^*(x)}, -N_p + \mu_p s, d_p) \prod_{q=1}^{Q} x_q^{-N'_q + \mu'_q s - 1} dx_q. \tag{2.16}
$$

*Proof.* Let $s = \sigma + i\tau \in \mathbb{C}$ such that $\sigma \gg_{\mathbf{N},\mathbf{N}'} 1$. By formula 2.15, we found that

$$
J_{\mathbf{N}, \mathbf{N}'}(\theta, s) = \frac{1}{\prod_{q=1}^{Q} \Gamma(-N'_q + \mu'_q s)} \sum_{\mathbf{n} \in \mathbb{N}_0^P} \prod_{p=1}^{P} (n_p + d_p)^{N_p - \mu_p s} \int_0^\theta \dots \int_0^\theta \prod_{p=1}^{P} e^{-(\sum_{q=1}^Q c_{q,p} x_q) n_p} \prod_{q=1}^Q e^{-d'_q x_q} x_q^{-N'_q + \mu'_q s - 1} dx_q.
$$
We know wish to interchange the integral sign with the sum. Firstly, we will bound the general term in the previous series. For all $q \in [1, Q]$, we define

$$v_q(\mathbf{n}) := \sum_{p=1}^{P} \mathrm{Re}(c_{q,p}) n_p.$$

We then found the following bound:

$$\forall \mathbf{n} \in \mathbb{N}_0^P, \quad \left| \prod_{p=1}^P (n_p + d_p)^{N_p - \mu_p s} \int_0^\theta \dots \int_0^\theta \prod_{p=1}^P e^{-(\sum_{q=1}^Q c_{q,p} x_q) n_p} \prod_{q=1}^Q e^{-d'_q x_q} x_q^{-N'_q + \mu'_q s - 1} dx_1 \dots dx_Q \right| \\ \le \prod_{p=1}^P |n_p + d_p|^{N_p - \mu_p s} e^{\mu_p |\tau| \pi/2} \int_0^\theta \dots \int_0^\theta \prod_{q=1}^Q e^{-(v_q(\mathbf{n}) + \mathrm{Re}(d'_q)) x_q} x_q^{-N'_q + \mu'_q \sigma - 1} dx_1 \dots dx_Q.$$

By a change of variables $y_q := (v_q(\mathbf{n}) + \mathrm{Re}(d'_q))x_q$ for all $q \in [1, Q]$, we get:

$$\left| \prod_{p=1}^{P} (n_p + d_p)^{N_p - \mu_p s} \int_0^\theta \dots \int_0^\theta \prod_{p=1}^P e^{-(\sum_{q=1}^Q c_{q,p} x_q) n_p} \prod_{q=1}^Q e^{-d'_q x_q} x_q^{-N'_q + \mu'_q s - 1} dx_1 \dots dx_Q \right| \\ \le \prod_{p=1}^P |n_p + d_p|^{N_p - \mu_p s} e^{\mu_p |\tau| \pi/2} \prod_{q=1}^Q (v_q(\mathbf{n}) + \mathrm{Re}(d'_q))^{N'_q - \mu'_q \sigma} \\ \cdot \int_0^{(v_1(\mathbf{n}) + \mathrm{Re}(d'_1))\theta} \dots \int_0^{(v_Q(\mathbf{n}) + \mathrm{Re}(d'_Q))\theta} \prod_{p=1}^P e^{-y_q} \prod_{q=1}^Q y_q^{-N'_q + \mu'_q \sigma - 1} dy_1 \dots dy_Q \\ \le \prod_{p=1}^P |n_p + d_p|^{N_p - \mu_p s} e^{\mu_p |\tau| \pi/2} \prod_{q=1}^Q (v_q(\mathbf{n}) + \mathrm{Re}(d'_q))^{N'_q - \mu'_q \sigma} \Gamma(-N'_q + \mu'_q \sigma).$$

We can see that the series $\sum_{\mathbf{n} \in \mathbb{N}_0^P} \prod_{p=1}^P |n_p + d_p|^{N_p - \mu_p s} e^{\mu_p |\tau| \pi/2} \prod_{q=1}^Q (v_q(\mathbf{n}) + \mathrm{Re}(d'_q))^{N'_q - \mu'_q \sigma}$ is convergent for $\sigma \gg_{\mu, \mu'} 1$ by Proposition 2.1.1. By Lebesgue's Theorem, for all $s = \sigma + i\tau \in \mathbb{C}$ such that $\sigma \gg_{\mu, \mu'} 1$, we get that

$$J_{\mathbf{N},\mathbf{N}'}(\theta, s) = \frac{1}{\prod_{q=1}^{Q} \Gamma(-N'_{q} + \mu'_{q}s)} \int_{0}^{\theta} \dots \int_{0}^{\theta} \prod_{p=1}^{P} e^{-l_{p}^{*}(x) d_{p}} \left( \sum_{n_{p}=1}^{+\infty} (n_{p} + d_{p})^{N_{p}-\mu_{p}s} e^{-l_{p}^{*}(x) n_{p}} \right) \prod_{q=1}^{Q} x_{q}^{-N'_{q} + \mu'_{q}s - 1} dx_{1} \dots dx_{Q},$$

which concludes the proof. $\square$

Once this integral formula is obtained for $J_{\mathbf{N},\mathbf{N}'}(\theta, s)$, we can use Erdélyi's Formula on each Lerch zeta function in the integrand's product, and then distribute the expression to get the following proposition:

**Proposition 2.3.20.** Let $(\mathbf{N}, \mathbf{N}') \in \mathbb{N}_0^P \times \mathbb{N}_0^Q$, and $(\mu, \mu') \in \mathbb{R}_+^P \times \mathbb{R}_+^Q$. Let $s \in \mathbb{C}$ such that $\mu_p s \notin \mathbb{N}$
for all $p \in [1, P]$, and $\sigma \gg 1$. Then, we find that for $\theta > 0$ sufficiently small,

$$
\begin{align}
J_{\mathbf{N},\mathbf{N}'}(\theta, s) = & \frac{1}{\prod_{q=1}^{Q} \Gamma(-N'_q + \mu'_q s)} \sum_{P \subseteq [1,P]} \prod_{p \in P} \Gamma(1+N_p - \mu_p s) \sum_{\mathbf{k}=(k_p)_{p \in P^c} \in \mathbb{N}_0^{P^c}} (-1)^{|\mathbf{k}|} \prod_{p \in P^c} \frac{\zeta(-N_p + \mu_p s - k_p, d_p)}{k_p!} \nonumber \\
& \cdot \int_0^\theta \dots \int_0^\theta \prod_{q=1}^Q x_q^{-N'_q + \mu'_q s - 1} \prod_{p \in P} l_p^*(x)^{-N_p + \mu_p s - 1} \prod_{p \in P^c} l_p^*(x)^{k_p} dx_1 \dots dx_Q \nonumber \\
& + \frac{\prod_{p=1}^P \Gamma(1+N_p - \mu_p s)}{\prod_{q=1}^Q \Gamma(-N'_q + \mu'_q s)} \int_0^\theta \dots \int_0^\theta \prod_{q=1}^Q x_q^{-N'_q + \mu'_q s - 1} \prod_{p=1}^P l_p^*(x)^{-N_p + \mu_p s - 1} dx_1 \dots dx_Q. \tag{2.17}
\end{align}
$$

Proof. Let $M = \max_{1 \le p \le P, 1 \le q \le Q} (|c_{q,p}|)$, and $\theta \in [0, \frac{1}{QM}]$. Let $s \in \mathbb{C}$ such that $\mu_p s \notin \mathbb{N}$ for all $p \in [1, P]$, and $\sigma \gg 1$. By the formula 2.16, we have that

$$
J_{\mathbf{N},\mathbf{N}'}(\theta, s) = \int_0^\theta \dots \int_0^\theta \sum_{\mathbf{n} \in \mathbb{N}_0^P} \prod_{p=1}^P e^{-d_p l_p^*(x)} \phi(e^{-l_p^*(x)}, -N_p + \mu_p s, d_p) \prod_{q=1}^Q x_q^{-N'_q + \mu'_q s - 1} dx_q.
$$

By Proposition 2.6, for all $(x_q)_{q \in [1, Q]} \in [0, \theta]^Q$, we have

$$
\begin{align*}
& \prod_{p=1}^{P} e^{-d_p l_p^*(x)} \phi(e^{-l_p^*(x)}, -N_p + \mu_p s, d_p) \prod_{q=1}^{Q} x_q^{-N'_q + \mu'_q s - 1} \\
&= \prod_{q=1}^{Q} x_q^{-N'_q + \mu'_q s - 1} \prod_{p=1}^{P} \left( \Gamma(1 + N_p - \mu_p s) l_p^*(x)^{-N_p + \mu_p s - 1} + \sum_{k_p \ge 0} (-1)^{k_p} \frac{l_p^*(x)^{k_p}}{k_p!} \zeta(-N_p + \mu_p s - k_p, d_p) \right).
\end{align*}
$$

Distributing the previous product, we get for all $\mathbf{x} = (x_q)_{q \in [1, Q]} \in [0, \theta]^Q$,

$$
\begin{align}
& \prod_{p=1}^{P} e^{-d_p l_p^*(x)} \phi(e^{-l_p^*(x)}, -N_p + \mu_p s, d_p) \prod_{q=1}^{Q} x_q^{-N'_q + \mu'_q s - 1} \nonumber \\
&= \prod_{q=1}^{Q} x_q^{-N'_q + \mu'_q s - 1} \sum_{P \subseteq [1, P]} \prod_{p \in P} \Gamma(1 + N_p - \mu_p s) \sum_{\mathbf{k}=(k_p)_{p \in P^c} \in \mathbb{N}_0^{P^c}} (-1)^{|\mathbf{k}|} \tag{2.18} \\
& \quad \cdot \prod_{p \in P} l_p^*(x)^{-N_p + \mu_p s - 1} \prod_{q=1}^{Q} l_q^*(x)^{k_q} \frac{\zeta(-N_p + \mu_p s - k_p, d_p)}{k_p!} \nonumber \\
& \quad + \prod_{q=1}^{Q} x_q^{-N'_q + \mu'_q s - 1} \prod_{p=1}^{P} \Gamma(1 + N_p - \mu_p s) \prod_{p=1}^{P} l_p^*(x)^{-N_p + \mu_p s - 1}. \nonumber
\end{align}
$$

Thus, we obtain

$$
\begin{align}
J_{\mathbf{N},\mathbf{N}'}(\theta, s) = & \frac{1}{\prod_{q=1}^{Q} \Gamma(-N'_q + \mu'_q s)} \sum_{P \subseteq [1,P]} \prod_{p \in P} \Gamma(1+N_p - \mu_p s) \int_0^\theta \dots \int_0^\theta \sum_{\mathbf{k}=(k_p)_{p \in P^c} \in \mathbb{N}_0^{P^c}} (-1)^{|\mathbf{k}|} \tag{2.19} \\
& \prod_{q=1}^{Q} x_q^{-N'_q + \mu'_q s - 1} \cdot \prod_{p \in P} l_p^*(x)^{-N_p + \mu_p s - 1} \prod_{q=1}^{Q} l_q^*(x)^{k_q} \frac{\zeta(-N_p + \mu_p s - k_p, d_p)}{k_p!} dx_1 \dots dx_Q \nonumber \\
& + \frac{\prod_{p=1}^{P} \Gamma(1+N_p - \mu_p s)}{\prod_{q=1}^{Q} \Gamma(-N'_q + \mu'_q s)} \int_0^\theta \dots \int_0^\theta \prod_{q=1}^{Q} x_q^{-N'_q + \mu'_q s - 1} \prod_{p=1}^{P} l_p^*(x)^{-N_p + \mu_p s - 1} dx_1 \dots dx_Q. \nonumber
\end{align}
$$
We shall now bound the general term in the series inside the integrand. By compactness, we have uniformly on $\mathbf{x} = (x_1, ..., x_Q) \in [0, \frac{1}{QM}]^Q$ that

$$ \left| \prod_{q=1}^{Q} x_q^{-N'_q+\mu'_q s-1} \right| \ll 1 \\ \left| l_p^*(x)^{-N_p+\mu_p s-1} \right| \ll 1. $$

Since $l_p^*$ is a linear form for all $p \in [1, P]$, we get that

$$ \forall x \in [0, \theta]^Q, \quad |l_p^*(x)| \le QM\theta. $$

By Lemma 2.3.2, we have the following bound:

$$ \forall p \in [1, P], \quad \left| \frac{\zeta(-N_p + \mu_p s - k_p, d_p)}{k_p!} \right| \ll \frac{(k_p + N_p)!}{k_p!} (k_p + N_p + 1)^{1+\lceil \mu_p |s| \rceil}. $$

For all $\mathcal{P} \subsetneq [1, P]$, and for all $\mathbf{x} = (x_q)_{q \in [1, Q]} \in [0, \theta]^Q$, we then get the following bound

$$ \left| \prod_{q=1}^{Q} x_q^{-N'_q+\mu'_q s-1} \prod_{p \in \mathcal{P}} l_p^*(x)^{-N_p+\mu_p s-1} \prod_{p \in \mathcal{P}^c} l_p^*(x)^{k_p} \frac{\zeta(-N_p + \mu_p s - k_p, d_p)}{k_p!} \right| \\ \ll \prod_{p \in \mathcal{P}^c} \frac{(k_p + N_p)!}{k_p!} (k_p + N_p + 1)^{1+\lceil \mu_p |s| \rceil} (QM\theta)^{k_p}. $$

Since the following series

$$ \sum_{\mathbf{k} \in \mathbb{N}_0^{\mathcal{P}^c}} \prod_{p \in \mathcal{P}^c} \frac{(k_p + N_p)!}{k_p!} (k_p + N_p + 1)^{1+\lceil \mu_p |s| \rceil} (QM\theta)^{k_p}, $$

is absolutely convergent on $\theta \in [0, \frac{1}{QM}]$, we then can interchange the series and the integral to prove the proposition. $\square$

**Proposition 2.3.21.** Let $(\mathbb{N}, \mathbb{N}') \in \mathbb{N}_0^P \times \mathbb{N}_0^Q$, and $(\mu, \mu') \in \mathbb{R}_+^P \times \mathbb{R}_+^Q$. Let $s \in \mathbb{C}$ such that $\mu_p s \notin \mathbb{N}$ for all $p \in [1, P]$, and $\sigma \gg 1$. We have

$$ J_{\mathbb{N}, \mathbb{N}'}(\theta, s) = \sum_{\mathcal{P} \subsetneq [1, P]} \sum_{j=1}^{Q} \left( \prod_{p \in \mathcal{P}} \Gamma(1 + N_p - \mu_p s) \right) \sum_{\mathbf{k}=(k_p)_{p \in \mathcal{P}^c} \in \mathbb{N}_0^{\mathcal{P}^c}} (-1)^{|\mathbf{k}|} h_{\mathbb{N}, \mathbb{N}', \mathcal{P}, j, \mathbf{k}}(s) \frac{\theta(|\mathbf{\mu}'|+|\mathbf{\mu}_{|\mathcal{P}}|)s - |\mathbb{N}'| - |\mathbb{N}_{|\mathcal{P}}| - |\mathcal{P}| + |\mathbf{k}|}{\Gamma(-N'_j + \mu'_j s)((|\mathbf{\mu}'| + |\mathbf{\mu}_{|\mathcal{P}}|)s - |\mathbb{N}'| - |\mathbb{N}_{|\mathcal{P}}| - |\mathcal{P}| + |\mathbf{k}|)} \prod_{p \in \mathcal{P}^c} \frac{\zeta(-N_p + \mu_p s - k_p, d_p)}{k_p!}, \\ + \left( \prod_{p=1}^{P} \Gamma(1 + N_p - \mu_p s) \right) \left( \sum_{j=1}^{Q} h_{\mathbb{N}, \mathbb{N}', [1, P], j}(s) \frac{\theta(|\mathbf{\mu}'|+|\mathbf{\mu}|)s - |\mathbb{N}'| - |\mathbb{N}| - P}{\Gamma(-N'_j + \mu'_j s)((|\mathbf{\mu}'| + |\mathbf{\mu}|)s - |\mathbb{N}'| - |\mathbb{N}| - P)} \right), $$

where we recall the notations of Proposition 2.2.4: For $\mathcal{P} \subsetneq [1, P]$, $j \in [1, Q]$, and $\mathbf{k} = (k_p)_{p \in \mathcal{P}^c} \in \mathbb{N}_0^{\mathcal{P}^c}$,

$$ h_{\mathbb{N}, \mathbb{N}', \mathcal{P}, j, \mathbf{k}}(s) = \frac{1}{\prod_{q=1}^{Q} \Gamma(-N'_q + \mu'_q s)} \cdot \int_{0}^{1} \dots \int_{0}^{1} \prod_{q \neq j}^{Q} x_q^{-N'_q + \mu'_q s - 1} \prod_{p \in \mathcal{P}} l_p^*(\hat{\mathbf{x}}^j)^{-N_p + \mu_p s - 1} \prod_{p \in \mathcal{P}^c} l_p^*(\hat{\mathbf{x}}^j)^{k_p} dx_1 \dots \widehat{dx_j} \dots dx_Q, $$
and

$$h_{\mathbf{N},\mathbf{N}',[1,P],j}(s) = \frac{1}{\prod_{q=1}^{Q} \Gamma(-N'_q + \mu'_q s)} \\ \cdot \int_{0}^{1} \dots \int_{0}^{1} \prod_{\substack{q=1 \\ q \neq j}}^{Q} x_q^{-N'_q + \mu'_q s - 1} \prod_{p=1}^{P} l_p^*(\hat{\mathbf{x}}^j)^{-N_p + \mu_p s - 1} dx_1 \dots \hat{dx}_j \dots dx_Q.$$

with:

$$1) \hat{\mathbf{x}}^j = (x_1, \dots, x_{j-1}, 1, x_{j+1}, \dots, x_Q),$$

$$2) dx_1 \dots \widehat{dx_j} \dots dx_Q = dx_1 \dots dx_{j-1} dx_{j+1} \dots dx_Q.$$

*Proof.* Let $s \in \mathbb{C}$ such that $\mu_p s \notin \mathbb{N}$ for all $p \in [1, P]$, and $\sigma \gg 1$. To prove the required formula, it's enough to revisit the formula obtained in the previous proposition, and to re-express the integral in this formula. This integral will be simplified using a blowing-up.

First, we note that $[0, \theta]^Q = \bigcup_{j=1}^Q V_j$, where $V_j := \{\mathbf{x} \in [0, \theta]^Q | \forall q \neq j, x_j \ge x_q\}$, and we also observe that the pairwise intersection of these sets has zero measure. By Chasles' relation, we obtain

$$\int_0^\theta \dots \int_0^\theta \prod_{q=1}^Q x_q^{-N'_q + \mu'_q s - 1} \prod_{p \in \mathcal{P}} l_p^*(\mathbf{x})^{-N_p + \mu_p s - 1} \prod_{p \in \mathcal{P}^c} l_p^*(\mathbf{x})^{k_p} dx_1 \dots dx_Q \\ = \sum_{j=1}^Q \int_{V_j} \prod_{q=1}^Q x_q^{-N'_q + \mu'_q s - 1} \prod_{p \in \mathcal{P}} l_p^*(\mathbf{x})^{-N_p + \mu_p s - 1} \prod_{p \in \mathcal{P}^c} l_p^*(\mathbf{x})^{k_p} dx_1 \dots dx_Q.$$

We now consider the following change of variables:

$$f_j : [0, 1]^{j-1} \times [0, \theta] \times [0, 1]^{Q-j} \to V_j \\ \mathbf{x} = (x_1, \dots, x_Q) \mapsto (x_1 x_j, \dots, x_j, \dots, x_Q x_j),$$

with the corresponding jacobian determinant $Jac(f_j) = x_j^{Q-1}$.

We can just study the integral with the domain $V_Q$, and extrapolate the other integrals by a permutation of variables. Firstly, let's note that for all $\mathbf{x} = (x_1, \dots, x_Q) \in [0, 1]^{Q-1} \times [0, \theta]$, we have $l_p^*(f_Q(\mathbf{x})) = x_Q l_p^*(\hat{\mathbf{x}}^Q)$. Therefore we get that

$$\int_{V_Q} \prod_{q=1}^{Q} x_q^{-N'_q + \mu'_q s - 1} \prod_{p \in \mathcal{P}} l_p^*(\mathbf{x})^{-N_p + \mu_p s - 1} \prod_{p \in \mathcal{P}^c} l_p^*(\mathbf{x})^{k_p} dx_1 \dots dx_Q \\ = \int_0^1 \dots \int_0^1 \int_0^\theta x_Q^{-|\mathbf{N}'| + |\mu'|s - Q - |\mathbf{N}|_P + |\mu|_P s - |\mathcal{P}| + |\mathbf{k}|} \prod_{q=1}^{Q-1} x_q^{-N'_q + \mu'_q s - 1} \\ \prod_{p \in \mathcal{P}} l_p^*(\hat{\mathbf{x}}^Q)^{-N_p + \mu_p s - 1} \prod_{p \in \mathcal{P}^c} l_p^*(\hat{\mathbf{x}}^Q)^{k_p} x_Q^{Q-1} dx_1 \dots dx_Q \\ = \int_0^1 \dots \int_0^1 \prod_{q=1}^{Q-1} x_q^{-N'_q + \mu'_q s - 1} \prod_{p \in \mathcal{P}} l_p^*(\hat{\mathbf{x}}^Q)^{-N_p + \mu_p s - 1} \prod_{p \in \mathcal{P}^c} u_p(\hat{\mathbf{x}}^Q)^{k_p} dx_1 \dots dx_{Q-1} \\ \cdot \frac{\theta(|\mu'| + |\mu|_P)s - |\mathbf{N}'| - |\mathbf{N}|_P - |\mathcal{P}| + |\mathbf{k}|}{(|\mu'| + |\mu|_P)s - |\mathbf{N}|' - |\mathbf{N}|_P - |\mathcal{P}| + |\mathbf{k}|}$$

which concludes the proof. $\square$

<figure><img src="image_16.png" /></figure>
The previous proposition is crucial to the proof of Proposition 2.2.4. We shall prove that $J_{\mathbf{N},\mathbf{N}'}^{\mu,\mu'}(\theta, s)$ is regular at $s = 0$ via the previous proposition. Particularly, we will use Proposition 2.2.2, which gives a bound for the functions $h_{\mathbf{N},\mathbf{N}'}^{\mu,\mu'}(\mathbf{P},\mathbf{j},\mathbf{k})(s)$, uniform in $s$ on all compact sets of $\mathbb{C}$. We shall prove Proposition 2.2.2 in the next sub-section.

### 2.3.3 Proof of Proposition 2.2.2

In this subsection, we will prove Proposition 2.2.2. Let us recall the notation $l_p^*(x) = \sum_{q=1}^Q c_{q,p}x_q$ for every $p \in [1, P]$ and every $x \in \mathbb{R}^Q$. We will establish a lemma before proving this proposition. For every $\epsilon \in [0, \frac{1}{2MQ}]$, for each subset $Q \subset [1, Q]\setminus\{j\}$, and for all $q \in [1, Q]\setminus\{j\}$, we set

$$ \Delta_{Q,j}(\epsilon, q) = \begin{cases} [0, \epsilon] & \text{if } q \in Q \\ [\epsilon, 1] & \text{if } q \in \mathbb{Q}^c \end{cases} $$

and we define $F_{Q,j}(\epsilon) := \prod_{\substack{q=1 \\ q \neq j}}^Q \Delta_{Q,j}(\epsilon, q)$. We then can partition the following integration domain:

$$ \{(x_1, \dots, x_{j-1}, x_{j+1}, \dots, x_Q) | \forall q \neq j, x_q \in [0, 1]\} = \bigcup_{Q \subset [1, Q]\setminus\{j\}} F_{Q,j}(\epsilon). $$

Via Chales' formula, we can write $h_{\mathbf{N},\mathbf{N}',\mathbf{P},j,\mathbf{k}}(s)$ and $h_{\mathbf{N},\mathbf{N}',[1,P],j}$ as sums of the following integrals:

$$ h_{\mathbf{N},\mathbf{N}',\mathbf{P},j,\mathbf{k}}(s) = \sum_{Q \subset [1,Q]\setminus\{j\}} I_{\mathbf{N},\mathbf{N}',\mathbf{P},j,Q,\mathbf{k}}(\epsilon, s) \quad (2.20) $$

$$ h_{\mathbf{N},\mathbf{N}',[1,P],j}(s) = \sum_{Q \subset [1,Q]\setminus\{j\}} I_{\mathbf{N},\mathbf{N}',[1,P],j,Q}(\epsilon, s), \quad (2.21) $$

where we noted for all subset $Q \subset [1, Q]\setminus\{j\}$,

$$ I_{\mathbf{N},\mathbf{N}',\mathbf{P},j,Q,\mathbf{k}}(\epsilon, s) = \frac{1}{\prod_{q \in [1,Q]\setminus\{j\}} \Gamma(-N'_q + \mu'_q s)} \int_{F_{Q,j}(\epsilon)} \prod_{\substack{q=1 \\ q \neq j}}^Q x_q^{-N'_q + \mu'_q s - 1} \prod_{p \in \mathcal{P}} l_p^*(\hat{\mathbf{x}}^j)^{-N_p + \mu_p s - 1} \prod_{p \in \mathcal{P}^c} l_p^*(\hat{\mathbf{x}}^j)^{k_p} dx_1 \dots \widehat{dx_j} \dots dx_Q, $$

and

$$ I_{\mathbf{N},\mathbf{N}',[1,P],j,Q}(\epsilon, s) = \frac{1}{\prod_{q \in [1,Q]\setminus\{j\}} \Gamma(-N'_q + \mu'_q s)} \int_{F_{Q,j}(\epsilon)} \prod_{\substack{q=1 \\ q \neq j}}^Q x_q^{-N'_q + \mu'_q s - 1} \prod_{p=1}^P l_p^*(\hat{\mathbf{x}}^j)^{-N_p + \mu_p s - 1} dx_1 \dots \widehat{dx_j} \dots dx_Q, $$

for all $s \in \mathbb{C}$ such that $\sigma \gg 1$.

**Lemma 2.3.22.** Let $(\mu, \mu') \in \mathbb{R}_+^P \times \mathbb{R}_+^Q$, $(\mathbf{N}, \mathbf{N}') \in \mathbb{N}_0^P \times \mathbb{N}_0^Q$, $\mathcal{P} \subset [1, P]$, $j \in [1, Q]$, $Q \subset [1, Q]\setminus\{j\}$, and $\mathbf{k} = (k_p)_{p \in \mathcal{P}} \in \mathbb{N}_0^{\mathcal{P}^c}$. Then we have that

i) For all $\epsilon > 0$ small enough, the function $I_{\mathbf{N},\mathbf{N}',\mathbf{P},j,Q,\mathbf{k}}(\epsilon, s)$ is holomorphic in $s$ on all $\mathbb{C}$.

ii) For $\epsilon > 0$ small enough, and for every positive integer $r \in \mathbb{N}$, we have the following bound, uniformly in $s \in D_0(r)$:

$$ \left| I_{\mathbf{N},\mathbf{N}',\mathbf{P},j,Q,\mathbf{k}}(\epsilon, s) \right| \ll (QM)^{|\mathbf{k}|}, $$
with $M = \max_{1 \le q \le Q, 1 \le p \le P} (|c_{q,p}|)$.
iii) For all $s \in \mathbb{C}$ and $0 < \epsilon \ll 1$, we have

$$
\begin{align}
I_{\mathbf{N},\mathbf{N}',\mathcal{P},j,\mathcal{Q},\mathbf{k}}(\epsilon, s) = & \sum_{\mathbf{v}=(\mathbf{v}_p)_{p \in \mathcal{P}} \in (\mathbb{N}_0^Q)^\mathcal{P}} \sum_{\substack{\mathbf{w}=(\mathbf{w}_p)_{p \in \mathcal{P}^c} \in (\mathbb{N}_0^Q)^\mathcal{P} \\ \forall p \in \mathcal{P}^c | \mathbf{w}_p| = k_p}} \left( \prod_{p \in \mathcal{P}^c} \binom{k_p}{\mathbf{w}_p} \prod_{q \in [1,Q]} c_{q,p}^{w_{p,q}} \right) \left( \prod_{p \in \mathcal{P}} \binom{-N_p + \mu_p s - 1}{|\mathbf{v}_p|} \binom{|\mathbf{v}_p|}{\mathbf{v}_p} \prod_{q \in \mathcal{Q}} c_{q,p}^{v_{p,q}} \right) \nonumber \\
& \cdot \frac{1}{\prod_{q \in [1,Q] \setminus \{j\}} \Gamma(-N'_q + \mu'_q s + \mathbf{v}(q) + \mathbf{w}(q))} \nonumber \\
& \cdot \int_{\epsilon}^{1} \dots \int_{\epsilon}^{1} \prod_{q \in \mathcal{Q}^c} x_q^{-N'_q + \mu'_q s - 1 + \mathbf{w}(q)} \prod_{p \in \mathcal{P}} \left( c_{j,p} + \sum_{f \in \mathcal{Q}^c} c_{f,p} x_f \right)^{-N_p + \mu_p s - 1 - |\mathbf{v}_p|} \prod_{q \in \mathcal{Q}^c} dx_q, \tag{2.22}
\end{align}
$$

where we set $\mathcal{Q}^c = (\llbracket 1, Q \rrbracket \setminus \{j\}) \setminus \mathcal{Q}$, $\mathbf{v}(q) = \sum_{p \in \mathcal{P}} v_{p,q}$ for all $q \in \mathcal{Q}$, and $\mathbf{w}(q) = \sum_{p \in \mathcal{P}^c} w_{p,q}$ for all $q \in \llbracket 1, Q \rrbracket$. Noting $T_{\mathbf{N},\mathbf{N}',\mathcal{P},j,\mathcal{Q},\mathbf{k},\mathbf{w},\mathbf{v}}(\epsilon, s)$ as the general term in the previous series, we also have

$$
\partial_s I_{\mathbf{N},\mathbf{N}',\mathcal{P},j,\mathcal{Q},\mathbf{k}}(\epsilon, s) = \sum_{\mathbf{v}=(\mathbf{v}_p)_{p \in \mathcal{P}} \in (\mathbb{N}_0^Q)^\mathcal{P}} \sum_{\substack{\mathbf{w}=(\mathbf{w}_p)_{p \in \mathcal{P}^c} \in (\mathbb{N}_0^Q)^\mathcal{P} \\ \forall p \in \mathcal{P}^c |\mathbf{w}_p|=k_p}} \partial_s T_{\mathbf{N},\mathbf{N}',\mathcal{P},j,\mathcal{Q},\mathbf{k},\mathbf{w},\mathbf{v}}(\epsilon, s) \quad (2.23)
$$

iv) For $\epsilon > 0$ small enough, and for $s \in \mathbb{C}$, we have

$$
h_{\mathbf{N},\mathbf{N}',\mathcal{P},j,\mathbf{k}}(s) = \sum_{\substack{\mathcal{Q} \subset [1,Q] \setminus \{j\} \\ \mu,\mu' \ge \mu}} I_{\mathbf{N},\mathbf{N}',\mathcal{P},j,\mathcal{Q},\mathbf{k}}(\epsilon, s) \quad (2.24)
$$

Proof. Here we aim to expand the integrand using power series expansions around 0, in $F_{Q,j}(\epsilon)$. Note that these power series will converge uniformly on the disk $D_0(\eta)$, for $\eta$ sufficiently small. We can therefore interchange the series and the integral.

Let $s \in \mathbb{C}$ such that $\sigma \gg 1$. Let $\mathcal{Q} \subset \llbracket 1, Q \rrbracket \setminus \{j\}$. Let $\epsilon \in \left[0, \frac{1}{2MQ}\right]$, with $M = \max_{1 \le q \le Q, 1 \le p \le P} (|c_{q,p}|)$, we can expand into a power series in $(x_q)_{q \in \mathcal{Q}}$ the functions of the form $l_p^*(\hat{x}^j)^{-N_p+\mu_p s-1}$, where $x \in F_{Q,j}$ and $p \in \mathcal{P}$. By Lemma 2.3.1, we obtain uniformly in $\hat{x}^j$:

$$
\begin{align}
l_p^*(\hat{\mathbf{x}}^j)^{-N_p+\mu_p s-1} &= \left( c_{j,p} + \sum_{q \in \mathcal{Q}^c} c_{q,p} x_q + \sum_{q \in \mathcal{Q}} c_{q,p} x_q \right)^{-N_p+\mu_p s-1} \nonumber \\
&= \left( c_{j,p} + \sum_{q \in \mathcal{Q}^c} c_{q,p} x_q \right)^{-N_p+\mu_p s-1} \left( 1 + \sum_{f \in \mathcal{Q}} \frac{c_{f,p}}{c_{j,p} + \sum_{q \in \mathcal{Q}^c} c_{q,p} x_q} x_f \right)^{-N_p+\mu_p s-1} \nonumber \\
&= \sum_{\mathbf{v}_p = (v_{p,q})_{q \in [1,Q]} \in \mathbb{N}_0^Q} \binom{-N_p + \mu_p s - 1}{|\mathbf{v}_p|} \binom{|\mathbf{v}_p|}{\mathbf{v}_p} \left( c_{j,p} + \sum_{q \in \mathcal{Q}^c} c_{q,p} x_q \right)^{-N_p+\mu_p s-1-|\mathbf{v}_p|} \prod_{q \in \mathcal{Q}} c_{q,p}^{v_{p,q}} x_q^{v_{p,q}}. \tag{2.25}
\end{align}
$$

For $p \in \mathcal{P}^c$, and $k_p \in \mathbb{N}_0$, we have by Newton's multinomial:

$$
l_p^*(\hat{\mathbf{x}}^j)^{k_p} = \sum_{\substack{\mathbf{w}_p=(w_{p,q})_{q \in [1,Q]} \in \mathbb{N}_0^Q \\ |\mathbf{w}_p|=k_p}} \binom{k_p}{\mathbf{w}_p} c_{j,p}^{w_{p,j}} \prod_{\substack{q=1 \\ q \neq j}}^{Q} c_{q,p}^{w_{p,q}} x_q^{w_{p,q}}.
$$
The series in (2.25) converges uniformly in $(x_e)_{e \in Q} \in [0, \epsilon]^Q$. By interchanging the integral and the series, we get that $I_{\mathbf{N},\mathbf{N}',\mathcal{P},j,\mathcal{Q},\mathbf{k}}(\epsilon, s)$ is of the following form:

$$ I_{\mathbf{N},\mathbf{N}',\mathcal{P},j,\mathcal{Q},\mathbf{k}}(\epsilon, s) = \sum_{\mathbf{v}=(\mathbf{v}_p)_{p \in \mathcal{P}} \in (\mathbb{N}_0^Q)^\mathcal{P}} \sum_{\substack{\mathbf{w}=(\mathbf{w}_p)_{p \in \mathcal{P}^c} \in (\mathbb{N}_0^Q)^{\mathcal{P}^c} \\ \forall p \in \mathcal{P}^c | \mathbf{w}_p| = k_p}} \left( \prod_{p \in \mathcal{P}^c} \binom{k_p}{\mathbf{w}_p} c_{j,p}^{w_{p,j}} \right) \left( \prod_{p \in \mathcal{P}} \binom{-N_p + \mu_p s - 1}{|\mathbf{v}_p|} \binom{|\mathbf{v}_p|}{|\mathbf{v}_p|} \right) \cdot \frac{1}{\prod_{q \in [1, Q] \setminus \{j\}} \Gamma(-N'_q + \mu'_q s)} \int_{F_{\mathcal{Q},j}(\epsilon)} \prod_{\substack{q=1 \\ q \neq j}}^Q x_q^{-N'_q + \mu'_q s - 1} \left( \prod_{\substack{p \in \mathcal{P}^c \\ q=1}}^Q \prod_{\substack{q=1 \\ q \neq j}}^Q c_{q,p}^{w_{p,q}} x_q^{w_{p,q}} \right) \cdot \prod_{p \in \mathcal{P}} \left( \prod_{q \in \mathcal{Q}} c_{q,p}^{v_{p,q}} x_q^{v_{p,q}} \right) \left( c_{j,p} + \sum_{q \in \mathcal{Q}^c} c_{q,p} x_q \right)^{-N_p + \mu_p s - 1 - |\mathbf{v}_p|} dx_1 \dots \widehat{dx_j} \dots dx_Q $$

Let's note that $[1, Q] \setminus \{j\} = \mathcal{Q} \cup \mathcal{Q}^c$. Then we can split the products $\prod_{q \in [1, Q] \setminus \{j\}}$ into 2 parts, and we get:

$$ I_{\mathbf{N},\mathbf{N}',\mathcal{P},j,\mathcal{Q},\mathbf{k}}(\epsilon, s) = \sum_{\mathbf{v}=(\mathbf{v}_p)_{p \in \mathcal{P}} \in (\mathbb{N}_0^Q)^\mathcal{P}} \sum_{\substack{\mathbf{w}=(\mathbf{w}_p)_{p \in \mathcal{P}^c} \in (\mathbb{N}_0^Q)^{\mathcal{P}^c} \\ \forall p \in \mathcal{P}^c | \mathbf{w}_p| = k_p}} \left( \prod_{p \in \mathcal{P}^c} \binom{k_p}{\mathbf{w}_p} \prod_{q=1}^Q c_{q,p}^{w_{p,q}} \right) \left( \prod_{p \in \mathcal{P}} \binom{-N_p + \mu_p s - 1}{|\mathbf{v}_p|} \binom{|\mathbf{v}_p|}{|\mathbf{v}_p|} \prod_{q \in \mathcal{Q}} c_{q,p}^{v_{p,q}} \right) \cdot \frac{1}{\prod_{q \in [1, Q] \setminus \{j\}} \Gamma(-N'_q + \mu'_q s)} \int_{F_{\mathcal{Q},j}(\epsilon)} \prod_{p \in \mathcal{P}^c} \left( \prod_{\substack{q=1 \\ q \neq j}}^Q x_q^{v_{p,q}} \right) \prod_{p \in \mathcal{P}} \left( \prod_{q \in \mathcal{Q}} x_q^{v_{p,q}} \right) \cdot \left( c_{j,p} + \sum_{q \in \mathcal{Q}^c} c_{q,p} x_q \right)^{-N_p + \mu_p s - 1 - |\mathbf{v}_p|} \cdot \prod_{q=1}^Q x_q^{-N'_q + \mu'_q s - 1} \cdot \prod_{q \neq j}^Q x_q^{-N'_q + \mu'_q s - 1} dx_1 \dots \widehat{dx_j} \dots dx_Q \\ = \sum_{\mathbf{v}=(\mathbf{v}_p)_{p \in \mathcal{P}} \in (\mathbb{N}_0^Q)^\mathcal{P}} \sum_{\substack{\mathbf{w}=(\mathbf{w}_p)_{p \in \mathcal{P}^c} \in (\mathbb{N}_0^Q)^{\mathcal{P}^c} \\ \forall p \in \mathcal{P}^c | \mathbf{w}_p| = k_p}} \left( \prod_{p \in \mathcal{P}^c} \binom{k_p}{\mathbf{w}_p} \prod_{q=1}^Q c_{q,p}^{w_{p,q}} \right) \left( \prod_{p \in \mathcal{P}} \binom{-N_p + \mu_p s - 1}{|\mathbf{v}_p|} \binom{|\mathbf{v}_p|}{|\mathbf{v}_p|} \prod_{q \in \mathcal{Q}} c_{q,p}^{v_{p,q}} \right) \cdot \frac{1}{\prod_{q \in [1, Q] \setminus \{j\}} \Gamma(-N'_q + \mu'_q s)} \int_{F_{\mathcal{Q},j}(\epsilon)} \prod_{q \in \mathcal{Q}} x_q^{-N'_q + \mu'_q s - 1 + \sum_{p \in \mathcal{P}} v_{p,q} + \sum_{p \in \mathcal{P}^c} w_{p,q}} \prod_{q \in \mathcal{Q}^c} x_q^{-N'_q + \mu'_q s - 1 + \sum_{p \in \mathcal{P}^c} w_{p,q}} \cdot \prod_{p \in \mathcal{P}} \left( c_{j,p} + \sum_{q \in \mathcal{Q}^c} c_{q,p} x_q \right)^{-N_p + \mu_p s - 1 - |\mathbf{v}_p|} dx_1 \dots \widehat{dx_j} \dots dx_Q $$

Let $\mathbf{v} = (\mathbf{v}_p)_{p \in \mathcal{P}}$ with $\mathbf{v}_p \in \mathbb{N}_0^Q$ for all $p \in \mathcal{P}$, and $\mathbf{w} = (\mathbf{w}_p)_{p \in \mathcal{P}^c}$ with $\mathbf{w}_p \in \mathbb{N}_0^Q$ for all $p \in \mathcal{P}^c$. We recall the notation $\mathbf{v}(q) = \sum_{p \in \mathcal{P}} v_{p,q}$ for all $q \in \mathcal{Q}$, and $\mathbf{w}(q) = \sum_{p \in \mathcal{P}^c} w_{p,q}$ for all $q \in [1, Q]$. Via
Fubini's theorem, we get the following expression for all $\sigma \gg 1$:

$$
\begin{align*}
I_{\mathbf{N},\mathbf{N}',\mathcal{P},j,\mathcal{Q},\mathbf{k}}(\epsilon, s) = & \sum_{\mu,\mu'} \sum_{\substack{\mathbf{v}=(\mathbf{v}_p)_{p \in \mathcal{P}} \in (\mathbb{N}_0^Q)^\mathcal{P} \\ \forall p \in \mathcal{P}^c | \mathbf{w}_p| = k_p}} \left( \prod_{p \in \mathcal{P}^c} \binom{k_p}{\mathbf{w}_p} \prod_{q=1}^Q c_{q,p}^{w_{p,q}} \right) \left( \prod_{p \in \mathcal{P}} \binom{-N_p + \mu_p s - 1}{|\mathbf{v}_p|} \binom{|\mathbf{v}_p|}{\mathbf{v}_p} \prod_{q \in \mathcal{Q}} c_{q,p}^{v_{p,q}} \right) \right. \\
& \left. \cdot \frac{1}{\prod_{q \in [1,Q] \setminus \{j\}} \Gamma(-N'_q + \mu'_q s)} \left( \prod_{q \in \mathcal{Q}} \frac{\epsilon^{-N'_q + \mu'_q s + \mathbf{v}(q) + \mathbf{w}(q)}}{-N'_q + \mu'_q s + \mathbf{v}(q) + \mathbf{w}(q)} \right) \int_{\epsilon}^{1} \dots \int_{\epsilon}^{1} \prod_{q \in \mathcal{Q}^c} x_q^{-N'_q + \mu'_q s - 1 + \mathbf{w}(q)} \right. \\
& \left. \cdot \prod_{p \in \mathcal{P}} \left( c_{j,p} + \sum_{q \in \mathcal{Q}^c} c_{q,p} x_q \right)^{-N_p + \mu_p s - 1 - |\mathbf{v}_p|} \prod_{q \in \mathcal{Q}^c} dx_q \right).
\end{align*}
$$

Thus we obtain that 2.22 is true for all $s \in \mathbb{C}$ such that $\sigma \gg 1$.

We're now looking to prove an analytic continuation of $s \mapsto I_{\mathbf{N},\mathbf{N}',\mathcal{P},j,\mathcal{Q},\mathbf{k}}(\epsilon, s)$ on all $\mathbb{C}$ via the formula that we just obtained. We clearly have that, for all $\sigma \gg 1$,

$$
\begin{align}
I_{\mathbf{N},\mathbf{N}',\mathcal{P},j,\mathcal{Q},\mathbf{k}}(\epsilon, s) = & \frac{1}{\prod_{q \in \mathcal{Q}^c} \Gamma(-N'_q + \mu'_q s)} \sum_{\mathbf{v}=(\mathbf{v}_p)_{p \in \mathcal{P}} \in (\mathbb{N}_0^Q)^\mathcal{P}} \sum_{\substack{\mathbf{w}=(\mathbf{w}_p)_{p \in \mathcal{P}^c} \in (\mathbb{N}_0^Q)^\mathcal{P} \\ \forall p \in \mathcal{P}^c |\mathbf{w}_p|=k_p}} \left( \prod_{p \in \mathcal{P}^c} \binom{k_p}{\mathbf{w}_p} \prod_{q=1}^Q c_{q,p}^{w_{p,q}} \right) \nonumber \\
& \left( \prod_{p \in \mathcal{P}} \binom{-N_p + \mu_p s - 1}{|\mathbf{v}_p|} \binom{|\mathbf{v}_p|}{\mathbf{v}_p} \prod_{q \in \mathcal{Q}} c_{q,p}^{v_{p,q}} \right) \prod_{q \in \mathcal{Q}} \frac{\epsilon^{-N'_q + \mu'_q s + \mathbf{v}(q) + \mathbf{w}(q)}}{\Gamma(-N'_q + \mu'_q s)(-N'_q + \mu'_q s + \mathbf{v}(q) + \mathbf{w}(q))} \nonumber \\
& \cdot \int_{\epsilon}^{1} \dots \int_{\epsilon}^{1} \left( \prod_{q \in \mathcal{Q}^c} x_q^{-N'_q + \mu'_q s - 1 + \mathbf{w}(q)} \right) \prod_{p \in \mathcal{P}} \left( c_{j,p} + \sum_{q \in \mathcal{Q}^c} c_{q,p} x_q \right)^{-N_p + \mu_p s - 1 - |\mathbf{v}_p|} \prod_{q \in \mathcal{Q}^c} dx_q. \tag{2.26}
\end{align}
$$

Let's define $T_{\mathbf{N},\mathbf{N}',\mathcal{P},j,\mathcal{Q},\mathbf{k},\mathbf{v},\mathbf{w}}(\epsilon, s)$ as the general term of the previous series. We then have that

$$
I_{\mathbf{N},\mathbf{N}',\mathcal{P},j,\mathcal{Q},\mathbf{k}}(\epsilon, s) = \sum_{\mu,\mu'} \sum_{\substack{\mathbf{v}=(\mathbf{v}_p)_{p \in \mathcal{P}} \in (\mathbb{N}_0^Q)^\mathcal{P} \\ \forall p \in \mathcal{P}^c |\mathbf{w}_p|=k_p}} T_{\mathbf{N},\mathbf{N}',\mathcal{P},j,\mathcal{Q},\mathbf{k},\mathbf{v},\mathbf{w}}(\epsilon, s).
$$

All the terms inside the expression of $T_{\mathbf{N},\mathbf{N}',\mathcal{P},j,\mathcal{Q},\mathbf{k},\mathbf{v},\mathbf{w}}(\epsilon, s)$ are holomorphic in $s$ on all $\mathbb{C}$. In order to show that $I_{\mathbf{N},\mathbf{N}',\mathcal{P},j,\mathcal{Q},\mathbf{k}}(\epsilon, s)$ is holomorphic in $s$, it's enough to show that the last series converges normally on all compact sets of $\mathbb{C}$. We're now looking to find a bound of that general term over a compact disk $\bar{D}_0(r)$, with $r \in \mathbb{N}$ being a positive integer. Such bound will depend on $\mathbf{N}, \mathbf{N}', \mu, \mu', \mathcal{P}, j, \mathcal{Q}$, but we shall not specify this dependence to increase the clarity of the proof.

We know that the function

$$s \mapsto \frac{1}{\Gamma(-N'_{q} + \mu'_{q}s)(-N'_{q} + \mu'_{q}s + \mathbf{v}(q) + \mathbf{w}(q))}$$

is holomorphic on $\mathbb{C}$, for all $q \in \mathcal{Q}$. We now bound the general term of the series 2.26 thanks to Lemma 2.3.4. We have

$$
\begin{align*}
& \forall q \in \mathcal{Q}, \forall s \in D_0(r), \forall a \in \mathbb{N}_0, \\
& \qquad \left| \frac{1}{\Gamma(-N'_{q} + \mu'_{q}s)(-N'_{q} + \mu'_{q}s + a)} \right| \ll 1 \\
& \forall q \in [1, Q], \qquad \left| \frac{1}{\Gamma(-N'_{q} + \mu'_{q}s)} \right| \ll 1.
\end{align*}
$$
We know that the function

$$s \in \mathbb{C} \mapsto \int_{\epsilon}^{1} \dots \int_{\epsilon}^{1} \prod_{q \in \mathbb{Q}^c} x_q^{-N'_q + \mu'_q s - 1 + \mathbf{w}(q)} \cdot \prod_{p \in \mathcal{P}} \left( c_{j,p} + \sum_{q \in \mathbb{Q}^c} c_{q,p} x_q \right)^{-N_p + \mu_p s - 1 - |\mathbf{v}_p|} \prod_{q \in \mathbb{Q}^c} dx_q,$$

is holomorphic because the integrand is regular.

Let $s \in D_0(r)$, $\mathbf{v} = (\mathbf{v}_p)_{p \in \mathcal{P}}$ with $\mathbf{v}_p \in \mathbb{N}^Q$, and $\mathbf{w} = (\mathbf{w}_p)_{p \in \mathcal{P}^c}$ with $\mathbf{w}_p \in \mathbb{N}_0^Q$ such that $|\mathbf{w}_p| = k_p$ for all $p \in \mathcal{P}^c$. We see that $\sum_{\substack{q=1 \\ q \neq j}}^Q \mathbf{w}(q) = |\mathbf{k}| - \mathbf{w}(j)$. Moreover, by compactness, we find the following bound which is uniform in $x_q \in [\epsilon, 1]$,

$$|x_q^{-N'_q + \mu'_q s - 1}| \ll 1.$$

As $\mathbf{w}(q) \ge 0$, we find that $|x_q^{\mathbf{w}(q)}| \le 1$. Moreover, since $\arg \left( c_{j,p} + \sum_{q \in \mathbb{Q}^c} c_{q,p} x_q \right) \in [-\frac{\pi}{2}, \frac{\pi}{2}]$, we have for all $s \in D_0(r)$, and for all $(x_q)_{q \in \mathbb{Q}^c} \in [\epsilon, 1]^{\mathbb{Q}^c}$,

$$\forall s \in D_0(r), \quad \left| \left( c_{j,p} + \sum_{q \in \mathbb{Q}^c} c_{q,p} x_q \right)^{-N_p + \mu_p s - 1 - |\mathbf{v}_p|} \right| \le \left| c_{j,p} + \sum_{q \in \mathbb{Q}^c} c_{q,p} x_q \right|^{-N_p + \mu_p \sigma - 1 - |\mathbf{v}_p|} e^{\mu_p r \pi / 2}.$$

Let's note that we have the following upper bound for all $(x_q)_{q \in \mathbb{Q}^c} \in [\epsilon, 1]^{\mathbb{Q}^c}$,

$$K + K\epsilon \le \left( \operatorname{Re}(c_{j,p} + \sum_{q \in \mathbb{Q}^c} \operatorname{Re}(c_{q,p}\epsilon)) \right) \le \left| c_{j,p} + \sum_{q \in \mathbb{Q}^c} c_{q,p} x_q \right| \le \sum_{q=1}^{Q} |c_{q,p}| \le QM,$$

with $M = \max_{1 \le p \le P, 1 \le q \le Q} (|c_{q,p}|)$ and $K := \min_{1 \le q \le Q, 1 \le p \le P} \operatorname{Re}(c_{q,p})$. For $p \in \mathcal{P}$, we then find that

$$\left| c_{j,p} + \sum_{q \in \mathbb{Q}^c} c_{q,p} x_q \right|^{-N_p + \mu_p \sigma - 1 - |\mathbf{v}_p|} \le \begin{cases} (QM)^{\mu_p \sigma - N_p - |\mathbf{v}_p| - 1} & \text{if } \left| c_{j,p} + \sum_{q \in \mathbb{Q}^c} c_{q,p} x_q \right| \ge 1 \text{ and } \mu_p \sigma - N_p - |\mathbf{v}_p| - 1 \ge 0 \\ 1 & \text{if } \left| c_{j,p} + \sum_{q \in \mathbb{Q}^c} c_{q,p} x_q \right| \ge 1 \text{ and } \mu_p \sigma - N_p - |\mathbf{v}_p| - 1 < 0 \\ 1 & \text{if } \left| c_{j,p} + \sum_{q \in \mathbb{Q}^c} c_{q,p} x_q \right| < 1 \text{ and } \mu_p \sigma - N_p - |\mathbf{v}_p| - 1 \ge 0 \\ (K(\epsilon+1))^{\mu_p \sigma - N_p - |\mathbf{v}_p| - 1} & \text{otherwise} \end{cases}$$

We have uniformly in $s \in D_0(r)$:

$$\forall p \in \mathcal{P}, \quad (QM)^{\mu_p \sigma - N_p - |\mathbf{v}_p| - 1} \ll (QM)^{-|\mathbf{v}_p|}.$$

Let $p \in [1, P]$. By compactness, we have uniformly in $s \in D_0(r)$:

$$(K\epsilon)^{\mu_p \sigma} \ll 1.$$

We also have

$$(K(\epsilon + 1))^{-N_p - |\mathbf{v}_p| - 1} \ll (K(\epsilon + 1))^{-|\mathbf{v}_p|} \ll K^{-|\mathbf{v}_p|}.$$
Finally, we have the following bound uniformly in $s \in D_0(r)$ and in $(x_q)_{q \in \mathbb{Q}^c} \in [\epsilon, 1]^{\mathbb{Q}^c}$,

$$ \left| \left( c_{j,p} + \sum_{q \in \mathbb{Q}^c} c_{q,p} x_q \right)^{-N_p + \mu_p \sigma - 1 - |\mathbf{v}_p|} \right| \ll \max_{r, \epsilon} \left( K^{-|\mathbf{v}_p|}, (QM)^{-|\mathbf{v}_p|}, 1 \right) \prod_{p=1}^{P} e^{\mu_p r \pi / 2} \ll \min_{r, \epsilon} (K, QM, 1)^{-|\mathbf{v}_p|}. $$

Now that we have a bound of the integrand, we can give a bound of the corresponding integral:

$$ \left| \int_{\epsilon}^{1} \dots \int_{\epsilon}^{1} \prod_{q \in \mathbb{Q}^c} x_q^{-N'_q + \mu'_q s - 1 + \mathbf{w}(q)} \prod_{p \in \mathcal{P}} \left( c_{j,p} + \sum_{q \in \mathbb{Q}^c} c_{q,p} x_q \right)^{-N_p + \mu_p s - 1 - |\mathbf{v}_p|} \prod_{q \in \mathbb{Q}^c} dx_q \right| \ll \prod_{r, \epsilon} \min(K, QM, 1)^{-|\mathbf{v}_p|} \ll \min_{r, \epsilon} (K, QM, 1)^{-\sum_{p \in \mathcal{P}} |\mathbf{v}_p|}. $$

We have the following bound, uniformly in $s \in D_0(r)$:

$$ \left| \epsilon^{\sum_{q \in \mathbb{Q}} (-N'_q + \mu'_q s + \mathbf{v}(q) + \mathbf{w}(q))} \right| \ll \epsilon^{\sum_{q \in \mathbb{Q}} \mathbf{v}(q)}. $$

Moreover, we have

$$ \forall n \in \mathbb{N}_0, \forall a \in \mathbb{N}_0, \forall b \in \mathbb{R}^*_+, \forall s \in D_0(r), \quad \left| \binom{-n+bs-1}{a} \right| \le \binom{n+\lceil br \rceil + a}{a}. $$

We then find that

$$ \begin{aligned} & \left| \frac{1}{\prod_{q \in \mathbb{Q}^c} \Gamma(-N'_q + \mu'_q s)} \left( \prod_{p \in \mathcal{P}^c} \binom{k_p}{w_p} \prod_{q=1}^Q c_{q,p}^{w_p,q} \right) \left( \prod_{p \in \mathcal{P}} \binom{-N_p + \mu_p s - 1}{|\mathbf{v}_p|} \binom{|\mathbf{v}_p|}{\mathbf{v}_p} \prod_{q \in \mathbb{Q}} c_{q,p}^{v_p,q} \right) \right. \\ & \qquad \left. \cdot \prod_{q \in \mathbb{Q}} \frac{\epsilon^{-N'_q + \mu'_q s + \mathbf{v}(q) + \mathbf{w}(q)}}{\Gamma(-N'_q + \mu'_q s)(-N'_q + \mu'_q s + \mathbf{v}(q) + \mathbf{w}(q))} \right. \\ & \qquad \left. \cdot \int_{\epsilon}^{1} \dots \int_{\epsilon}^{1} \prod_{q \in \mathbb{Q}^c} x_q^{-N'_q + \mu'_q s - 1 + \mathbf{w}(q)} \cdot \prod_{p \in \mathcal{P}} \left( c_{j,p} + \sum_{q \in \mathbb{Q}^c} c_{q,p} x_q \right)^{-N_p + \mu_p s - 1 - |\mathbf{v}_p|} \prod_{q \in \mathbb{Q}^c} dx_q \right| \\ & \ll \left( \prod_{r, \epsilon} \binom{k_p}{w_p} M^{\sum_{q \in \mathbb{Q}} w_{p,q}} \prod_{p \in \mathcal{P}} \binom{N_p + \lceil \mu_p r \rceil + |\mathbf{v}_p|}{|\mathbf{v}_p|} \binom{|\mathbf{v}_p|}{\mathbf{v}_p} \right) (M\epsilon)^{\sum_{q \in \mathbb{Q}} \mathbf{v}(q)} \min(K, QM, 1)^{-\sum_{q \in \mathbb{Q}} \mathbf{v}(q)} \\ & \ll \left( \prod_{r, \epsilon} \binom{k_p}{w_p} M^{k_p} \prod_{p \in \mathcal{P}} \binom{N_p + \lceil \mu_p r \rceil + |\mathbf{v}_p|}{\mathbf{v}_p} \right) \left( \frac{M\epsilon}{\min(K, QM, 1)} \right)^{\sum_{q \in \mathbb{Q}} \mathbf{v}(q)}. \end{aligned} \quad (2.27) $$

Taking the sum of the right hand side of 2.27 over $\mathbf{v} = (\mathbf{v}_p)_{p \in \mathcal{P}}$ with $\mathbf{v}_p \in \mathbb{N}^{\mathbb{Q}}$, and over $\mathbf{w} = (\mathbf{w}_p)_{p \in \mathcal{P}^c}$ with $\mathbf{w}_p \in \mathbb{N}_0^Q$ such that $|\mathbf{w}_p| = k_p$ for all $p \in \mathcal{P}^c$, we get

$$ M^{|\mathbf{k}|} \left( \sum_{\substack{\mathbf{w}=(\mathbf{w}_p)_{p \in \mathcal{P}^c} \in (\mathbb{N}_0^Q)^{\mathcal{P}^c} \\ \forall p \in \mathcal{P}^c |\mathbf{w}_p|=k_p}} \prod_{p \in \mathcal{P}^c} \binom{k_p}{w_p} \right) \left( \sum_{\substack{\mathbf{v}=(\mathbf{v}_p)_{p \in \mathcal{P}} \in (\mathbb{N}_0^Q)^{\mathcal{P}} \\ \forall p \in \mathcal{P}^c |\mathbf{v}_p|=k_p}} \prod_{p \in \mathcal{P}} \binom{N_p + \lceil \mu_p r \rceil + |\mathbf{v}_p|}{\mathbf{v}_p} \right) \left( \frac{M\epsilon}{\min(K, QM, 1)} \right)^{\sum_{q \in \mathbb{Q}} \mathbf{v}(q)} \\ = (QM)^{|\mathbf{k}|} \left( \sum_{\substack{\mathbf{v}=(\mathbf{v}_p)_{p \in \mathcal{P}} \in (\mathbb{N}_0^Q)^{\mathcal{P}} \\ \forall p \in \mathcal{P}^c |\mathbf{v}_p|=k_p}} \prod_{p \in \mathcal{P}} \binom{N_p + \lceil \mu_p r \rceil + |\mathbf{v}_p|}{\mathbf{v}_p} \right) \left( \frac{M\epsilon}{\min(K, QM, 1)} \right)^{\sum_{q \in \mathbb{Q}} \mathbf{v}(q)}. $$
The previous power series converges absolutely on $0 < \epsilon < \frac{\min(K, QM, 1)}{M}$. By taking a $\epsilon > 0$ small enough, we then find that the previous series over $(v_{p,q})_{p \in \mathcal{P}, q \in \mathcal{Q}}$ is a constant depending only on $r, \epsilon$ and on $\mathbf{N}$. We then get the following inequality:

$$ \left| I_{\mathbf{N}, \mathbf{N}', \mathcal{P}, j, \mathcal{Q}, \mathbf{k}}(\epsilon, s) \right| \le \frac{(QM)^{|\mathbf{k}|}}{r \cdot \epsilon} $$

Thus we get i) and iii) by the Lebesgue's theorem, and by analytic continuation. ii) is true by the above upper bound. As $I_{\mathbf{N}, \mathbf{N}', \mathcal{P}, j, \mathcal{Q}, \mathbf{k}}(\epsilon, s)$ is holomorphic, and because for all $s = \sigma + i\tau \in \mathbb{C}$ such that $\sigma \gg 1$, we have

$$ h_{\mathbf{N}, \mathbf{N}', \mathcal{P}, j, \mathbf{k}}(s) = \sum_{\mathcal{Q} \subset [[1, Q]] \setminus \{j\}} I_{\mathbf{N}, \mathbf{N}', \mathcal{P}, j, \mathcal{Q}, \mathbf{k}}(\epsilon, s), $$

by 2.20, we obtain that the previous equality stands on all $\mathbb{C}$ by analytic continuation, thus proving iv). $\square$

By a similar reasoning, we get the following lemma:

**Lemma 2.3.23.** Let $(\mu, \mu') \in \mathbb{R}_+^P \times \mathbb{R}_+^Q$, and $(\mathbf{N}, \mathbf{N}') \in \mathbb{N}_0^P \times \mathbb{N}_0^Q$, $j \in [[1, Q]]$, $\mathcal{Q} \subset [[1, Q]] \setminus \{j\}$. For all $s \in \mathbb{C}$ and $0 < \epsilon \ll 1$, we have:

$$ I_{\mathbf{N}, \mathbf{N}', [[1, P]], j, \mathcal{Q}}(\epsilon, s) = \sum_{\mathbf{v}=(\mathbf{v}_p)_{p \in [[1, P]]} \in (\mathbb{N}_0^Q)^P} \left( \prod_{q \in [[1, Q]]} c_{q,p}^{w_{p,q}} \right) \left( \prod_{p=1}^P \left( \frac{-N_p + \mu_p s - 1}{|\mathbf{v}_p|} \right) \left( \frac{|\mathbf{v}_p|}{\mathbf{v}_p} \right) \prod_{q \in \mathcal{Q}} c_{q,p}^{v_{p,q}} \right) \cdot \frac{1}{\prod_{q \in [[1, Q]] \setminus \{j\}} \Gamma(-N'_q + \mu'_q s + \mathbf{v}(q))} \cdot \int_{\epsilon}^{1} \dots \int_{\epsilon}^{1} \prod_{q \in \mathcal{Q}^c} x_q^{-N'_q + \mu'_q s - 1} \cdot \prod_{p=1}^{P} \left( c_{j,p} + \sum_{f \in \mathcal{Q}^c} c_{f,p} x_f \right)^{-N_p + \mu_p s - 1 - |\mathbf{v}_p|} \prod_{q \in \mathcal{Q}^c} dx_q, $$

with the notations $\mathcal{Q}^c = ([[1, Q]] \setminus \{j\}) \setminus \mathcal{Q}$, $\mathbf{v}(q) = \sum_{p=1}^P v_{p,q}$ for all $q \in \mathcal{Q}$. Moreover, $s \in \mathbb{C} \mapsto I_{\mathbf{N}, \mathbf{N}', [[1, P]], j, \mathcal{Q}}(\epsilon, s)$ is holomorphic in $\mathbb{C}$.

**Remark 2.3.24.** As opposed to Lemma 2.3.22, we don't need a bound in Lemma 2.3.23, because there is no $\mathbf{k}$ inside the expression of $I_{\mathbf{N}, \mathbf{N}', [[1, P]], j, \mathcal{Q}}(\epsilon, s)$.

We know have all the ingredients required to prove Proposition 2.2.2:

*Proof.* Let's recall that $h_{\mathbf{N}, \mathbf{N}', \mathcal{P}, j, \mathbf{k}}(s)$ and $h_{\mathbf{N}, \mathbf{N}', [[1, P]], j}(s)$ have the following expression for all $s \in \mathbb{C}$ such that $\sigma \gg 1$:

$$ h_{\mathbf{N}, \mathbf{N}', \mathcal{P}, j, \mathbf{k}}(s) = \sum_{\mathcal{Q} \subset [[1, Q]] \setminus \{j\}} I_{\mathbf{N}, \mathbf{N}', \mathcal{P}, j, \mathcal{Q}, \mathbf{k}}(\epsilon, s), \\ h_{\mathbf{N}, \mathbf{N}', [[1, P]], j}(s) = \sum_{\mathcal{Q} \subset [[1, Q]] \setminus \{j\}} I_{\mathbf{N}, \mathbf{N}', [[1, P]], j, \mathcal{Q}}(\epsilon, s). $$

Since the functions $I_{\mathbf{N}, \mathbf{N}', \mathcal{P}, j, \mathcal{Q}, \mathbf{k}}(\epsilon, s)$ and $I_{\mathbf{N}, \mathbf{N}', [[1, P]], j, \mathcal{Q}}(\epsilon, s)$ are holomorphic in $s$ on $\mathbb{C}$, we have an analytic continuation of $h_{\mathbf{N}, \mathbf{N}', \mathcal{P}, j, \mathbf{k}}(s)$ and of $h_{\mathbf{N}, \mathbf{N}', [[1, P]], j}(s)$ on all $s \in \mathbb{C}$. We then obtain point i) of Proposition 2.2.2. Point ii) and iii) of Proposition 2.2.2 follows directly from point ii) and iii) of Lemma 2.3.22. $\square$
### 2.3.4 Proof of Proposition 2.2.4

Let $(\mathbf{N}, \mathbf{N}') \in \mathbb{N}_0^P \times \mathbb{N}_0^Q$, and $(\boldsymbol{\mu}, \boldsymbol{\mu}') \in \mathbb{R}_+^P \times \mathbb{R}_+^Q$. We exploit the expression $Z(\mathbf{s}, \mathbf{s}') = K(\theta, \mathbf{s}, \mathbf{s}') + J(\theta, \mathbf{s}, \mathbf{s}')$ to obtain the proof of the meromorphic extension theorem in the direction $(\boldsymbol{\mu}, \boldsymbol{\mu}')$. We already know that $(\mathbf{s}, \mathbf{s}') \mapsto K(\theta, \mathbf{s}, \mathbf{s}')$ is holomorphic on $\mathbb{C}^P \times \mathbb{C}^Q$. What remains is to prove that the function $s \mapsto J_{\mathbf{N},\mathbf{N}'}(\theta, s)$ is meromorphic and regular at $s = 0$. In order to show Proposition 2.2.4, all that remains is to establish that $s \mapsto J_{\mathbf{N},\mathbf{N}'}(\theta, s)$ is meromorphic on $\mathbb{C}$ with poles contained in

$$S(\mathbf{N}, \mathbf{N}') = \left[ \bigcup_{\substack{1 \le p \le P \\ \mu_p \ne 0}} \frac{1}{\mu_p} \mathbb{Z}_{\ge N_p+1} \right] \left[ \bigcup_{\mathcal{P} \subset [1, P]} \left[ \bigcup_{\substack{|\boldsymbol{\mu}'| + |\boldsymbol{\mu}|_P \le \mathbb{Z}^* \\ |\boldsymbol{\mu}|_P \le |\mathbf{N}_P| + |\mathbf{N}'| + |\mathcal{P}|}} \frac{1}{|\boldsymbol{\mu}'| + |\boldsymbol{\mu}|_P} \right] \right].$$

We know that formula 2.5 is valid on the complement of this set, for $\sigma \gg 1$. For all $s = \sigma + i\tau \in \mathbb{C}\backslash S(\mathbf{N}, \mathbf{N}')$ such that $\sigma \gg 1$, we have that $\mu, \mu'$

$$J_{\mathbf{N},\mathbf{N}'}(\theta, s) = \sum_{\mathcal{P} \subset [1,P]} \left( \prod_{p \in \mathcal{P}} \Gamma(1+N_p - \mu_p s) \right) \sum_{\mathbf{k}=(k_p)_{p \in \mathcal{P}} \in \mathbb{N}_0^{\mathcal{P}^c}} (-1)^{|\mathbf{k}|} \left( \sum_{j=1}^Q h_{\mathbf{N},\mathbf{N}',\mathcal{P},j,\mathbf{k}}(s) \frac{\theta(|\boldsymbol{\mu}'|+|\boldsymbol{\mu}|_P)s - |\mathbf{N}'|-|\mathbf{N}_P|-|\mathcal{P}|+|\mathbf{k}|}{\Gamma(-N'_j + \mu'_j s)((|\boldsymbol{\mu}'|+|\boldsymbol{\mu}|_P)s - |\mathbf{N}'|-|\mathbf{N}_P|-|\mathcal{P}|+|\mathbf{k}|)} \right) \prod_{p \in \mathcal{P}^c} \frac{\zeta(-N_p + \mu_p s - k_p, d_p)}{k_p!} + \left( \prod_{p=1}^P \Gamma(1+N_p - \mu_p s) \right) \left( \sum_{j=1}^Q h_{\mathbf{N},\mathbf{N}',[1,P],j}(s) \frac{\theta(|\boldsymbol{\mu}'|+|\boldsymbol{\mu}|_P)s - |\mathbf{N}'|-|\mathbf{N}_P|-P}{\Gamma(-N'_j + \mu'_j s)((|\boldsymbol{\mu}'|+|\boldsymbol{\mu}|_P)s - |\mathbf{N}'|-|\mathbf{N}_P|-P)} \right).$$

Via all the above, we can now prove Proposition 2.2.4:

*Proof.* Let $0 < \theta \ll 1$. We know from Proposition 2.3.21 that the formula 2.5 is true for all $s = \sigma + i\tau \in \mathbb{C}\backslash S(\mathbf{N}, \mathbf{N}')$, such that $\sigma \gg 1$.

It is clear that

$$\sum_{j=1}^{Q} h_{\mathbf{N}, \mathbf{N}', [1, P], j}(s) \frac{\theta(|\boldsymbol{\mu}'|+|\boldsymbol{\mu}|)s - |\mathbf{N}'| - |\mathbf{N}_P| - P}{\Gamma(-N'_j + \mu'_j s)((|\boldsymbol{\mu}'|+|\boldsymbol{\mu}|)s - |\mathbf{N}'| - |\mathbf{N}_P| - P)},$$

is meromorphic in $s$ on all $\mathbb{C}$, and regular at $s = 0$, with poles of the form $s = \frac{|\mathbf{N}'| + |\mathbf{N}_P| + P}{|\boldsymbol{\mu}'| + |\boldsymbol{\mu}|}$. Then, it remains to show that the series of functions in $s$

$$\sum_{\mathcal{P} \subset [1, P]} \left( \prod_{p \in \mathcal{P}} \Gamma(1 + N_p - \mu_p s) \right) \sum_{\mathbf{k} = (k_p)_{p \in \mathcal{P}} \in \mathbb{N}_0^{\mathcal{P}^c}} (-1)^{|\mathbf{k}|} \left( \sum_{j=1}^{Q} h_{\mathbf{N}, \mathbf{N}', \mathcal{P}, j, \mathbf{k}}(s) \frac{\theta(|\boldsymbol{\mu}'| + |\boldsymbol{\mu}|_P)s - |\mathbf{N}'| - |\mathbf{N}_P| - |\mathcal{P}| + |\mathbf{k}|}{\Gamma(-N'_j + \mu'_j s)((|\boldsymbol{\mu}'| + |\boldsymbol{\mu}|_P)s - |\mathbf{N}'| - |\mathbf{N}_P| - |\mathcal{P}| + |\mathbf{k}|)} \right) \prod_{p \in \mathcal{P}^c} \frac{\zeta(-N_p + \mu_p s - k_p, d_p)}{k_p!} \quad (2.28)$$

is meromorphic on $\mathbb{C}$, regular at $s = 0$, and with poles inside $S(\mathbf{N}, \mathbf{N}')$.
We define

$$
\tilde{D}(\mathbf{N}, \mathbf{N}', \delta, r) := \\
\left\{
  \begin{array}{@{}l@{}}
    s \in \bar{D}_0(r) \left( \forall p \in [1, P], \min_{\substack{n \in \mathbb{N}_0 \\ n > N_p}} |\mu_p s - n| \ge \delta \right) \wedge \left( \forall \mathcal{P} \subset [1, P], \min_{\substack{n \in \mathbb{Z}^* \\ n \ge -|\mathbf{N}_{||\mathcal{P}||} - |\mathbf{N}'|| - |\mathcal{P}||}} |(|\boldsymbol{\mu}|_{\mathcal{P}} + |\boldsymbol{\mu}'|)s + n| \ge \delta \right) \right\}.
  \end{array}
\right.
$$

Let's note that

1) We clearly have that $\tilde{D}(\mathbf{N}, \mathbf{N}', \delta, r)$ contains an open neighbourhood of 0.

2) The condition

$$
\min_{\substack{n \in \mathbb{N}_0 \\ n > N_p}} |\mu_p s - n| \ge \delta
$$

is here to avoid the singularities of $\zeta(-N_p + \mu_p s - k, d_p)$. We see that, around $s = 0$, this last term is regular.

3) The condition

$$
\min_{\substack{n \in \mathbb{Z}^* \\ n \ge -|\mathbf{N}_{||\mathcal{P}||} - |\mathbf{N}'|| - |\mathcal{P}||}} |(|\boldsymbol{\mu}|_{\mathcal{P}} + |\boldsymbol{\mu}'|)s + n| \ge \delta
$$

is here to avoid the singularities of $\frac{1}{(|\boldsymbol{\mu}'| + |\boldsymbol{\mu}|_{\mathcal{P}})s - |\mathbf{N}'| - |\mathbf{N}_{||\mathcal{P}||} - |\mathcal{P}|| + |\mathbf{k}|}$ which are not necessarily compensated by the zeros of $\frac{1}{\Gamma(-N'_j + \mu'_j s)}$. We shall note that around $s = 0$, when $-|\mathbf{N}'| - |\mathbf{N}_{||\mathcal{P}||} - |\mathcal{P}|| + |\mathbf{k}| = 0$, the singularity of $\frac{1}{(|\boldsymbol{\mu}'| + |\boldsymbol{\mu}|_{\mathcal{P}})s}$ is compensated by the zero of $\frac{1}{\Gamma(-N'_j + \mu'_j s)}$.

4) The set $S(\mathbf{N}, \mathbf{N}')$ is inside the complement of $\tilde{D}(\mathbf{N}, \mathbf{N}', \delta, r)$ in $\mathbb{C}$.
$\mathbf{\mu}, \mathbf{\mu}'$

Let's fix a subset $\mathcal{P} \subsetneq [1, P]$. We see that by Proposition 2.3.21, the formula 2.5 is true for all
$s = \sigma + i\tau \in \mathbb{C} \setminus S(\mathbf{N}, \mathbf{N}')$, such that $\sigma \gg 1$. We wish to prove that the series in $\mathbf{k}$ in formula (2.28)
converges normally on $\tilde{D}(\mathbf{N}, \mathbf{N}', \delta, r)$.
$\mathbf{\mu}, \mathbf{\mu}'$

Let $r' > 1$ be a real such that, for all $p \in [1, P]$, we have $\mathrm{Re}(d_p) \ge 1/r'$, and $|d_p| < r'$. Let $r > 0$, $1 > \delta > 0$. By Lemma 2.3.3, we have

$$
\forall p \in [1, P], \forall s \in \tilde{D}(\mathbf{N}, \mathbf{N}', \delta, r), \forall k_p \in \mathbb{N}_0, \zeta(-N_p + \mu_p s - k_p, d_p) \ll_{r, r', \delta} (k_p + N_p)!(k_p + N_p + 1)^{r+1} (1 + r'^{2k_p}).
$$

By Lemma 2.2.2, we have

$$
\forall s \in D_0(r), \forall \mathbf{k} = (k_p)_{p \in \mathcal{P}^c} \in \mathbb{N}_0^{\mathcal{P}^c}, \quad |h_{\mathbf{N}, \mathbf{N}', \mathcal{P}, j, \mathbf{k}}(s)| \ll_{r} (QM)^{|\mathbf{k}|}.
$$

Since the function $s \mapsto \Gamma(1 + N_p - \mu_p s)$ is holomorphic on $\tilde{D}(\mathbf{N}, \mathbf{N}', \delta, r)$ for all $1 \le p \le P$, we then find
the following bound

$$
\forall p \in [1, P], \forall s \in \tilde{D}(\mathbf{N}, \mathbf{N}', \delta, r), \quad |\Gamma(1 + N_p - \mu_p s)|_{r, \delta} \ll 1.
$$

By Lemma 2.3.4 we find the following bound

$$
\forall s \in \tilde{D}(\mathbf{N}, \mathbf{N}', \delta, r), \left| \frac{1}{\Gamma(-N_j' + \mu_j's)((|\boldsymbol{\mu}'| + |\boldsymbol{\mu}|_{\mathcal{P}})s - |\mathbf{N}'| - |\mathbf{N}_{||\mathcal{P}||} - |\mathcal{P}|| + |\mathbf{k}|)} \right| \ll 1.
$$
By compactness of $\bar{D}_0(r)$ we get

$$ \forall s \in \bar{D}_0(r), |\theta(|\boldsymbol{\mu}'|+|\boldsymbol{\mu}|_P)s - |\boldsymbol{N}'| - |\boldsymbol{N}|_P - |\mathcal{P}|| \ll 1. $$

Therefore, for all $s \in \tilde{D}(\mathbf{N}, \mathbf{N}', \delta, r)$, and for $\mathbf{k} = (k_p)_{p \in \mathcal{P}^c} \in \mathbb{N}_0^{\mathcal{P}^c}$, we find

$$ \begin{aligned} & \left| (-1)^{|\mathbf{k}|} h_{\mathbf{N}, \mathbf{N}', \mathcal{P}, j, \mathbf{k}}(s) \prod_{p \in \mathcal{P}} \Gamma(1 + N_p - \mu_p s) \right. \\ & \qquad \left. \frac{\theta(|\boldsymbol{\mu}'|+|\boldsymbol{\mu}|_P)s - |\boldsymbol{N}'| - |\boldsymbol{N}|_P - |\mathcal{P}| + |\mathbf{k}|}{\Gamma(-N'_j + \mu'_j s)((|\boldsymbol{\mu}'| + |\boldsymbol{\mu}|_P)s - |\boldsymbol{N}'| - |\boldsymbol{N}|_P - |\mathcal{P}| + |\mathbf{k}|)} \prod_{p \in \mathcal{P}^c} \frac{\zeta(-N_p + \mu_p s - k_p, d_p)}{k_p!} \right| \\ & \ll_{r, r', \delta} (QM\theta)^{|\mathbf{k}|} \prod_{p \in \mathcal{P}^c} \frac{(k_p + N_p)!}{k_p!} (k_p + N_p + 1)^{r+1} (1 + r'^{2k_p}) \\ & \ll_{r, r', \delta} (2r'^2QM\theta)^{|\mathbf{k}|} \prod_{p \in \mathcal{P}^c} \frac{(k_p + N_p)!}{k_p!} (k_p + N_p + 1)^{r+1}. \end{aligned} $$

Since the series

$$ \sum_{\mathbf{N} \in \mathbb{N}_0^{\mathcal{P}^c}} (2r'^2QM\theta)^{|\mathbf{k}|} \prod_{p \in \mathcal{P}^c} \frac{(k_p + N_p)!}{k_p!} (k_p + N_p + 1)^{r+1} $$

converges absolutely for all $\theta \in \mathbb{C}$ such that $|\theta| < \frac{1}{(2r'^2QM)}$, we find that formula 2.5 holds for all $s \in \tilde{D}(\mathbf{N}, \mathbf{N}', \delta, r)$. By holomorphy theorem under the sum sign, we get that the function $s \mapsto J_{\mathbf{N}, \mathbf{N}'}(\theta, s)$ is holomorphic on $\tilde{D}(\mathbf{N}, \mathbf{N}', \delta, r)$. As $\tilde{D}(\mathbf{N}, \mathbf{N}', \delta, r)$ contains an open neighbourhood of 0, we see that by the same holomorphy theorem under the sum sign, we can derive termwise in respect to $s$ around $s = 0$ the function $J_{\mathbf{N}, \mathbf{N}'}(\theta, s)$,

$$ \partial_s J_{\mathbf{N}, \mathbf{N}'}(\theta, s) = \tag{2.29} $$

$$ \begin{aligned} & \sum_{\mathcal{P} \subsetneq [1, P]} \sum_{j=1}^{Q} \sum_{\mathbf{k}=(k_p)_{p \in \mathcal{P}^c} \in \mathbb{N}_0^{\mathcal{P}^c}} \partial_s \left( \left( \prod_{p \in \mathcal{P}} \Gamma(1 + N_p - \mu_p s) \right) (-1)^{|\mathbf{k}|} h_{\mathbf{N}, \mathbf{N}', \mathcal{P}, j, \mathbf{k}}(s) \right. \\ & \qquad \left. \frac{\theta(|\boldsymbol{\mu}'|+|\boldsymbol{\mu}|_P)s - |\boldsymbol{N}'| - |\boldsymbol{N}|_P - |\mathcal{P}| + |\mathbf{k}|}{\Gamma(-N'_j + \mu'_j s)((|\boldsymbol{\mu}'| + |\boldsymbol{\mu}|_P)s - |\boldsymbol{N}'| - |\boldsymbol{N}|_P - |\mathcal{P}| + |\mathbf{k}|)} \prod_{p \in \mathcal{P}^c} \frac{\zeta(-N_p + \mu_p s - k_p, d_p)}{k_p!} \right), \\ & + \partial_s \left( \prod_{p=1}^{P} \Gamma(1 + N_p - \mu_p s) \sum_{j=1}^{Q} h_{\mathbf{N}, \mathbf{N}', [1, P], j}(s) \frac{\theta(|\boldsymbol{\mu}'|+|\boldsymbol{\mu}|)s - |\boldsymbol{N}'| - |\boldsymbol{N}| - P}{\Gamma(-N'_j + \mu'_j s)((|\boldsymbol{\mu}'| + |\boldsymbol{\mu}|)s - |\boldsymbol{N}'| - |\boldsymbol{N}| - P)} \right). \end{aligned} $$

As $1 > \delta > 0$ is arbitrary in the definition of $\tilde{D}(\mathbf{N}, \mathbf{N}', \delta, r)$, and that $\tilde{D}(\mathbf{N}, \mathbf{N}', \delta, r)$ contains an open neighbourhood of $s = 0$, we then get that $J_{\mathbf{N}, \mathbf{N}'}(\theta, s)$ is regular at $s = 0$, and that the poles are inside $S(\mathbf{N}, \mathbf{N}')$ described in Proposition 2.2.4. $\square$

**Remark 2.3.25.** Via formula 2.5, one can find expressions of the residues of $Z_{\mathbf{N}, \mathbf{N}'}(s)$ at these poles. In fact, we know that this function can be decomposed into two sums, one one which is holomorphic on $\mathbb{C}$, and the other one is meromorphic:

$$ Z_{\mathbf{N}, \mathbf{N}'}(s) = K_{\mathbf{N}, \mathbf{N}'}(\theta, s) + J_{\mathbf{N}, \mathbf{N}'}(\theta, s). $$
Since $K_{\mathbf{N},\mathbf{N}'}(\theta, s)$ is holomorphic in $s$, this function doesn't have any singularities. On the other hand, in formula 2.5, one can see that compute the residues by studying carefully the two following terms for $j \in [1, Q]$,

$$ \bullet \frac{\theta(|\boldsymbol{\mu}'|+|\boldsymbol{\mu}|_P)s - |\boldsymbol{N}'|-|\boldsymbol{N}|_P - |\boldsymbol{P}| + |\boldsymbol{k}|}{\Gamma(-N'_j + \mu'_j s)((|\boldsymbol{\mu}'| + |\boldsymbol{\mu}|_P)s - |\boldsymbol{N}'| - |\boldsymbol{N}|_P - |\boldsymbol{P}| + |\boldsymbol{k}|)} \prod_{p \in \mathcal{P}^c} \frac{\zeta(-N_p + \mu_p s - k_p, d_p)}{k_p!} \quad (\mathcal{P} \subsetneq [1, P], \mathbf{k} \in \mathbb{N}_0^{\mathcal{P}^c}) $$

$$ \bullet \frac{\theta(|\boldsymbol{\mu}'|+|\boldsymbol{\mu}|)s - |\boldsymbol{N}'|-|\boldsymbol{N}|-P}{\Gamma(-N'_j + \mu'_j s)((|\boldsymbol{\mu}'| + |\boldsymbol{\mu}|)s - |\boldsymbol{N}'| - |\boldsymbol{N}|-P)}. $$

Let's note that these poles can sometimes be compensated by zeros of the functions $h_{\mathbf{N},\mathbf{N}',\mathcal{P},j,\mathbf{k}}(s)$ for the first term, and of zeros of the functions $h_{\mathbf{N},\mathbf{N}',[1,P],j}(s)$ for the second term.
<figure><img src="image_17.png" /></figure>
# Chapter 3

## Computation of the coefficients

$$Q_{\mathbf{N},\mathbf{N}'}^{0}(\mathcal{P}, j, \mathbf{k}) \text{ and } Q_{\mathbf{N},\mathbf{N}'}^{1}(\mathcal{P}, j, \mathbf{k})$$

$\mu, \mu'$

$\mu, \mu'$

By Proposition 2.2.2, we know that the function $h_{\mathbf{N},\mathbf{N}',\mathcal{P},j,\mathbf{k}}$ is holomorphic on $\mathbb{C}$. Therefore, we can consider his Taylor's coefficients of order 0 and 1 near $s = 0$:

**Notation.** Let $(\mu, \mu') \in \mathbb{R}^P \times \mathbb{R}_*^Q$, and $(\mathbf{N}, \mathbf{N}') \in \mathbb{N}_0^P \times \mathbb{N}_0^Q$. Let $\mathcal{P} \subset [1, P]$ and $\mathbf{k} \in \mathbb{N}_0^{P_c}$. As $h_{\mathbf{N},\mathbf{N}',\mathcal{P},j,\mathbf{k}}$ is holomorphic on $\mathbb{C}$, we consider the following coefficients:

$$h_{\mathbf{N},\mathbf{N}',\mathcal{P},j,\mathbf{k}}(s) = Q_{\mathbf{N},\mathbf{N}'}^{0}(\mathcal{P},j,\mathbf{k}) + Q_{\mathbf{N},\mathbf{N}'}^{1}(\mathcal{P},j,\mathbf{k})s + O(s^2).$$

The coefficients $Q_{\mathbf{N},\mathbf{N}'}^{0}(\mathcal{P},j,\mathbf{k})$, for all $j \in [1, Q]$, $\mathcal{P} \subsetneq [1, P]$, will be crucial in the expression of the formulas for $Z(-(\mathbf{N}, \mathbf{N}'))$. Furthermore, the coefficients $Q_{\mathbf{N},\mathbf{N}'}^{0}(\mathcal{P},j,\mathbf{k})$ and $Q_{\mathbf{N},\mathbf{N}'}^{1}(\mathcal{P},j,\mathbf{k})$, for all $j \in [1, Q]$, $\mathcal{P} \subsetneq [1, P]$, will appear in the formula for the directional derivative at nonpositive integers $Z'(-(\mathbf{N}, \mathbf{N}'))$. It is also worth noting that we do not need to compute $Q_{\mathbf{N},\mathbf{N}'}^{0}(\mathcal{P},j,\mathbf{k})$ and $Q_{\mathbf{N},\mathbf{N}'}^{1}(\mathcal{P},j,\mathbf{k})$, as they do not appear in the expressions contained in Theorem A and in Theorem D. We will show in our calculations that $Q_{\mathbf{N},\mathbf{N}'}^{0}(\mathcal{P},j,\mathbf{k})$ is independent of the directions $(\mu, \mu')$. We will therefore simply denote this coefficient as $Q_{\mathbf{N},\mathbf{N}'}^{0}(\mathcal{P},j,\mathbf{k})$.

### 3.1 Lemmata

**Lemma 3.1.1.** Let $a, c \in H_0$, $b \in \mathbb{N}_0$ and $d \in \mathbb{Z}$. We define $g: \mathbb{C} \to \mathbb{C}$ the following meromorphic function:

$$g(s) := \frac{1}{\Gamma(as - b)(cs + d)}.$$

Then $g$ is regular near $s = 0$. Moreover, we have:

$$g(0) = \begin{cases} \frac{a}{c}(-1)^b b! & \text{if } d = 0 \\ 0 & \text{if } d \neq 0, \end{cases}$$
and

$$ \partial_s (g(s))_{s=0} = \begin{cases} \frac{a^2}{c}(-1)^b b! (\gamma - h_b) & \text{if } d = 0 \\ \frac{a}{d}(-1)^b b! & \text{if } d \neq 0, \end{cases} $$

where $h_b$ is the harmonic number $h_b := \sum_{i=1}^{b} \frac{1}{i}$.

We also have

$$ \partial_s \left( \frac{1}{\Gamma(s)} \right) \Big|_{s=-N} = (-1)^N N!. $$

*Proof.* Around $s = 0$, we have

$$ \frac{1}{\Gamma(s-N)} = (-1)^N N! s + O(s^2), $$

which proves the formula for $\partial_s \left( \frac{1}{\Gamma(s)} \right)_{|s=-N}$.

In order to prove the other formulas, we shall study the behavior of $g$ near $s = 0$. By Gamma's functional equation, we get

$$ g(s) = \frac{(as - 1)...(as - b)}{\Gamma(as)(cs + d)}. $$

Around $s = 0$, we have

$$ \begin{aligned} (as - 1)...(as - b) &= (-1)^b b! + a(-1)^{b-1}b!h_b s + O(s^2), \\ \frac{1}{\Gamma(s)} &= s + \gamma s^2 + O(s^3). \end{aligned} $$

Let's assume that $d = 0$. Therefore, near $s = 0$ we have:

$$ \begin{aligned} g(s) &= \frac{a(as - 1)...(as - b)}{c \Gamma(as)as} \\ &= \frac{a}{c}((-1)^b b! + as(-1)^{b-1}b!h_b)(1 + \gamma as) + O(s^2) \\ &= \frac{a}{c}(-1)^b b! + \frac{a^2}{c}(-1)^{b-1}b!h_b s + \frac{a^2}{c}\gamma(-1)^b b!s + O(s^2). \end{aligned} $$

Let's assume that $d \neq 0$. Therefore, near $s = 0$ we have:

$$ \begin{aligned} g(s) &= \frac{(as - 1)...(as - b)}{\Gamma(as)(cs + d)} \\ &= \frac{(as - 1)...(as - b)}{cs + d}(as) + O(s^2) \\ &= \frac{a}{d}(-1)^b b! s + O(s^2). \end{aligned} $$

□

**Lemma 3.1.2.** Let $\mathcal{P} \subset [1, P]$, $(\mu_p)_{p \in \mathcal{P}} \in \mathbb{R}^{\mathcal{P}}$, $(N_p)_{p \in \mathcal{P}} \in \mathbb{N}_0^{\mathcal{P}}$, $(n_p)_{p \in \mathcal{P}} \in \mathbb{N}_0^{\mathcal{P}}$, we have

$$ \partial_s \left( \prod_{p \in \mathcal{P}} \binom{-N_p - 1 + \mu_p s}{n_p} \right) \Big|_{s=0} = \prod_{p \in \mathcal{P}} \binom{-N_p - 1}{n_p} \left( \sum_{p \in \mathcal{P}} \mu_p (h_{N_p} - h_{N_p + n_p}) \right), $$

where $h_n$ is the n-th harmonic number.
*Proof.* Let's recall that

$$ \prod_{p \in \mathcal{P}} \binom{-N_p - 1 + \mu_p s}{n_p} = \prod_{p \in \mathcal{P}} \frac{(-N_p - 1 + \mu_p s) \dots (-N_p + \mu_p s - n_p)}{n_p!} $$

By developing the product $\frac{(-N_p - 1 + \mu_p s) \dots (-N_p - n_p + \mu_p s)}{n_p!}$ near $s = 0$, we get

$$ \frac{(-N_p - 1 + \mu_p s) \dots (-N_p + \mu_p s - n_p)}{n_p!} = \binom{-N_p - 1}{n_p} + \mu_p \left( \sum_{i=0}^{n_p-1} \frac{(-N_p - 1) \dots \widehat{(-N_p - 1 - i)} \dots (-N_p - n_p)}{n_p!} \right) s + O(s^2), $$

where $\widehat{(-N_p - 1 - i)}$ means that we forget that term in the product. We can simplify that last expression:

$$ \begin{aligned} \frac{(-N_p - 1 + \mu_p s) \dots (-N_p + \mu_p s - n_p)}{n_p!} &= \binom{-N_p - 1}{n_p} + \mu_p \binom{-N_p - 1}{n_p} \left( \sum_{i=0}^{n_p-1} \frac{1}{-N_p - 1 - i} \right) s \\ &= \binom{-N_p - 1}{n_p} + \mu_p \binom{-N_p - 1}{n_p} (h_{N_p} - h_{N_p+n_p}) s + O(s^2). \end{aligned} $$

Near $s = 0$, we then get that:

$$ \prod_{p \in \mathcal{P}} \binom{-N_p - 1 + \mu_p s}{n_p} = \prod_{p \in \mathcal{P}} \binom{-N_p - 1}{n_p} + \prod_{p \in \mathcal{P}} \binom{-N_p - 1}{n_p} \left( \sum_{p \in \mathcal{P}} \mu_p (h_{N_p} - h_{N_p+n_p}) \right) s + O(s^2). $$

□

**Lemma 3.1.3.** Let $M \in \mathbb{Z}_{\le 0}$ be a nonpositive integer, and $(a_k)_{k \in \mathbb{I}_{M, +\infty[}$ and $(b_k)_{k \in \mathbb{I}_{M, +\infty[}$ be two complex number sequences such that the power series $\sum_{k=0}^{+\infty} a_k x^k$ and the power series $\sum_{k=0}^{+\infty} b_k x^k$ have two positive radius of convergence $R_1$ and $R_2$. Let's also assume that there exist a positive real number $0 < R < \min(R_1, R_2)$ such that

$$ \forall x \in ]0, R[, \quad \sum_{k=M}^{+\infty} a_k x^k + \sum_{k=M}^{+\infty} b_k x^k \ln(x) = 0, $$

Then for all $k \in \mathbb{I}]0, +\infty[$, we have $a_k = 0$ and $b_k = 0$.

*Proof.* If all $b_k$ are zeros (respectively all $a_k$ are zeros) for all $k \ge M$, the result is trivial. Otherwise, we shall use a reductio ad absurdum argument. Let's assume that there exist $i, j \ge M$ such that $a_i \ne 0$ and $b_j \ne 0$. Suppose that $i = \min\{k|a_k \ne 0\}$ and $j = \min\{k|b_k \ne 0\}$, then in the neighborhood of 0 we have

$$ a_i x^i + b_j x^j \ln(x) + o(x^j \ln(x)) + o(x^i) = 0, $$

which implies $b_j x^{j-i} \ln(x) \sim a_i$ near $x = 0$, a contradiction.

□

The preceding lemma, although trivial, will play a crucial role in determining formulas for special values in the proof of Theorem A and Theorem D. Indeed, in the decomposition of $Z_{(\mathbb{N}, \mathbb{N}')}(s)$ obtained in Proposition 2.2.4, we have artificially introduced a free variable $\theta$. It is clear that $K_{(\mathbb{N}, \mathbb{N}')}(\theta, s)$ vanishes at $s = 0$, the only challenge then is to evaluate $J_{(\mathbb{N}, \mathbb{N}')}(\theta, 0)$ using formula 2.5 at $s = 0$, which will yield an expression involving a Laurent series in $\theta$. Since the quantity $Z(-\mathbb{N}, \mathbb{N}') = Z_{(\mathbb{N}, \mathbb{N}')}(0)$ does not depend on $\theta$, we will only retain the constant terms of the Laurent series in $\theta$ from the obtained formula. For the
calculation of $Z'(-(\mathbf{N}, \mathbf{N}')) = Z'_{(\mathbf{N}, \mathbf{N}')(\mu, \mu')}(0)$, we will also use this lemma to make the same simplification, except that this time, by differentiating formula 2.5 with respect to $s$, a term involving $\ln(\theta)$ will appear. In the same way, we will use this lemma to compute the coefficients $Q^0$ and $Q^1$, in order to eliminate the series in $\epsilon$ and the multiples of $\ln(\epsilon)$.

**Lemma 3.1.4.** There exist a function defined for all $0 < \epsilon \ll 1$ such that

$$ \alpha_{\mathcal{P},j,f,\mathbf{v},\mathbf{w},\mathbf{N},\mathbf{N}'}^*(\epsilon) = a \ln(\epsilon) + \sum_{n \ge m} b_n \epsilon^n, $$

with $m \in \mathbb{Z}$, without constant terms (i.e. $b_0 = 0$), and such that

$$ \int_{\epsilon}^{1} x_f^{-N_f'-1+\mathbf{w}(f)} \cdot \prod_{p \in \mathcal{P}} (c_{j,p} + c_{f,p}x_f)^{-N_p+\mu_p s-1-|\mathbf{v}_p|} dx_f = \alpha_{\mathcal{P},j,f,\mathbf{v},\mathbf{w},\mathbf{N},\mathbf{N}'}^* (\epsilon) + F_{\mathcal{P},j,f,\mathbf{v},\mathbf{w}}(\mathbf{N},\mathbf{N}'), $$

with $F_{\mathcal{P},j,f,\mathbf{v},\mathbf{w}}(\mathbf{N},\mathbf{N}')$ being the constant defined in Proposition 1.3.2:

$$ F_{\mathcal{P},j,f,\mathbf{v},\mathbf{w}}(\mathbf{N},\mathbf{N}') = -E_{\mathcal{P},j,f,\mathbf{v},\mathbf{w}}(\mathbf{N},\mathbf{N}') - \sum_{\lambda=2}^{N_f'+1-\mathbf{w}(f)} \frac{C_{\mathcal{P},j,f,\mathbf{v},\mathbf{w},\lambda}(\mathbf{N},\mathbf{N}')}{\lambda-1} \\ - \frac{1}{c_{f,p}} \sum_{p \in \mathcal{P}} \sum_{\lambda=2}^{(|\mathbf{N}|_{\mathcal{P}}+|\mathcal{P}|+|\mathbf{v}|)} \frac{D_{\mathcal{P},j,f,\mathbf{v},\mathbf{w},(\lambda,p)}(\mathbf{N},\mathbf{N}')}{\lambda-1} ((c_{j,p} + c_{f,p})^{1-\lambda} - c_{j,p}^{1-\lambda}) \\ + \frac{1}{c_{f,p}} \sum_{p \in \mathcal{P}} D_{\mathcal{P},j,f,\mathbf{v},\mathbf{w},(1,p)}(\mathbf{N},\mathbf{N}') \ln \left(1 + \frac{c_{f,p}}{c_{j,p}}\right). $$

*Proof.* In order to compute the integral above, we shall use Proposition 1.3.2. With the same notations as in that proposition, we get

$$ x_f^{-N_f'-1+\mathbf{w}(f)} \cdot \prod_{p \in \mathcal{P}} (c_{j,p} + c_{f,p}x_f)^{(-N_p-1-|\mathbf{v}_p|)} = \tilde{P}_{\mathcal{P},j,f,\mathbf{v},\mathbf{w},\mathbf{N},\mathbf{N}'}(x_f) + \sum_{\lambda=1}^{N_f'+1-\mathbf{w}(f)} \frac{C_{\mathcal{P},j,f,\mathbf{v},\mathbf{w},\lambda}(\mathbf{N},\mathbf{N}')}{x_f^{\lambda}} \\ + \sum_{p \in \mathcal{P}} \sum_{\lambda=1}^{(|\mathbf{N}|_{\mathcal{P}}+|\mathcal{P}|+|\mathbf{v}|)} \frac{D_{\mathcal{P},j,f,\mathbf{v},\mathbf{w},(\lambda,p)}(\mathbf{N},\mathbf{N}')}{(c_{j,p} + c_{f,p}x_f)^{\lambda}}. $$

We noted $P_{\mathcal{P},j,f,\mathbf{v},\mathbf{w},\mathbf{N},\mathbf{N}'}(x_f)$ the primitive of the polynomial $\tilde{P}_{\mathcal{P},j,f,\mathbf{v},\mathbf{w},\mathbf{N},\mathbf{N}'}(x_f)$ which fades at $x_f = 1$, and we noted $E_{\mathcal{P},j,f,\mathbf{v},\mathbf{w}}(\mathbf{N},\mathbf{N}')$ the constant term of $P_{\mathcal{P},j,f,\mathbf{v},\mathbf{w},\mathbf{N},\mathbf{N}'}(x_f)$. We also noted $P_{\mathcal{P},j,f,\mathbf{v},\mathbf{w},\mathbf{N},\mathbf{N}'}^*(x_f) := P_{\mathcal{P},j,f,\mathbf{v},\mathbf{w},\mathbf{N},\mathbf{N}'}(x_f) - E_{\mathcal{P},j,f,\mathbf{v},\mathbf{w}}(\mathbf{N},\mathbf{N}')$. Therefore we find that

$$ \int_{\epsilon}^{1} x_f^{-N_f'-1+\mathbf{w}(f)} \cdot \prod_{p \in \mathcal{P}} (c_{j,p} + c_{f,p}x_f)^{-N_p+\mu_p s-1-|\mathbf{v}_p|} dx_f = \\ - E_{\mathcal{P},j,f,\mathbf{v},\mathbf{w}}(\mathbf{N},\mathbf{N}') - P_{\mathcal{P},j,f,\mathbf{v},\mathbf{w},\mathbf{N},\mathbf{N}'}^*(\epsilon) - C_{\mathcal{P},j,f,\mathbf{v},\mathbf{w},1}(\mathbf{N},\mathbf{N}') \ln(\epsilon) - \sum_{\lambda=2}^{N_f'+1-\mathbf{w}(f)} \frac{C_{\mathcal{P},j,f,\mathbf{v},\mathbf{w},\lambda}(\mathbf{N},\mathbf{N}')}{\lambda-1} (1-\epsilon^{1-\lambda}) \\ + \frac{1}{c_{f,p}} \sum_{p \in \mathcal{P}} D_{\mathcal{P},j,f,\mathbf{v},\mathbf{w},(1,p)}(\mathbf{N},\mathbf{N}') \left( \ln(c_{j,p} + c_{f,p}) - \ln(c_{j,p}) - \ln\left(1 + \frac{c_{f,p}}{c_{j,p}}\epsilon\right) \right) \\ - \frac{1}{c_{f,p}} \sum_{p \in \mathcal{P}} \sum_{\lambda=2}^{(|\mathbf{N}|_{\mathcal{P}}+|\mathcal{P}|+|\mathbf{v}|)} \frac{D_{\mathcal{P},j,f,\mathbf{v},\mathbf{w},(\lambda,p)}(\mathbf{N},\mathbf{N}')}{\lambda-1} ((c_{j,p} + c_{f,p})^{1-\lambda} - (c_{j,p} + c_{f,p}\epsilon)^{1-\lambda}). $$

By expanding the terms $(c_{j,p} + c_{f,p}\epsilon)^{1-\lambda}$ and $\ln\left(1 + \frac{c_{f,p}}{c_{j,p}}\epsilon\right)$ near $\epsilon = 0$, we can decompose the previously obtained formula into a constant term, and into a sum between a term in $\ln(\epsilon)$ and a Laurent series
without a constant term in $\epsilon$, for $0 < \epsilon \ll 1$. Let's denote the non-constant term $\alpha_{\mathcal{P},j,f,\mathbf{v},\mathbf{w},\mathbf{N},\mathbf{N}'}^*(\epsilon)$. We then have

$$ \begin{aligned} \int_{\epsilon}^{1} x_f^{-N_f'-1+\mathbf{w}(f)} \cdot \prod_{p \in \mathcal{P}} (c_{j,p} + c_{f,p}x_f)^{-N_p+\mu_p s-1-|\mathbf{v}_p|} dx_f = & \alpha_{\mathcal{P},j,f,\mathbf{v},\mathbf{w},\mathbf{N},\mathbf{N}'}^*(\epsilon) \\ & - E_{\mathcal{P},j,f,\mathbf{v},\mathbf{w}}(\mathbf{N},\mathbf{N}') - \sum_{\lambda=2}^{N_f'+1-\mathbf{w}(f)} \frac{C_{\mathcal{P},j,f,\mathbf{v},\mathbf{w},\lambda}(\mathbf{N},\mathbf{N}')}{\lambda-1} + \frac{1}{c_{f,p}} \sum_{p \in \mathcal{P}} D_{\mathcal{P},j,f,\mathbf{v},\mathbf{w},(1,p)}(\mathbf{N},\mathbf{N}') \ln \left(1 + \frac{c_{f,p}}{c_{j,p}}\right) \\ & - \frac{1}{c_{f,p}} \sum_{p \in \mathcal{P}} (|\mathbf{N}|_p + |\mathcal{P}| + |\mathbf{v}|) \frac{D_{\mathcal{P},j,f,\mathbf{v},\mathbf{w},(\lambda,p)}(\mathbf{N},\mathbf{N}')}{\lambda-1} \left((c_{j,p} + c_{f,p})^{1-\lambda} - c_{j,p}^{1-\lambda}\right). \end{aligned} $$

□

We note that a simpler, although less explicit, expression of the previous integral can be obtained, using a symbolic computation software:

Consider the primitive $G_{\mathcal{P},j,f,\mathbf{v},\mathbf{w},\mathbf{N},\mathbf{N}'}(x_f)$ of $x_f^{-N_f'-1+\mathbf{w}(f)} \prod_{p \in \mathcal{P}} (c_{j,p} + c_{f,p}x_f)^{-N_p+\mu_p s-1-|\mathbf{v}_p|}$ that vanishes at $x_f = 1$. This primitive is expressed as a sum of a multiple of $\ln(x_f)$ and a Laurent series in $x_f$, for $0 < x_f \ll 1$. We then observe that the constant term of this Laurent series is $-F_{\mathcal{P},j,f,\mathbf{v},\mathbf{w}}(\mathbf{N},\mathbf{N}')$.

**Remark 3.1.5.** It seems more practical to directly compute the coefficient $F_{\mathcal{P},j,f,\mathbf{v},\mathbf{w}}(\mathbf{N},\mathbf{N}')$ using symbolic computation software by determining a primitive of $x_f^{-N_f'-1+\mathbf{w}(f)} \prod_{p \in \mathcal{P}} (c_{j,p} + c_{f,p}x_f)^{-N_p+\mu_p s-1-|\mathbf{v}_p|}$, rather than computing the coefficients $C_{\mathcal{P},j,f,\mathbf{v},\mathbf{w},\lambda}(\mathbf{N},\mathbf{N}')$, $D_{\mathcal{P},j,f,\mathbf{v},\mathbf{w},(\lambda,p)}(\mathbf{N},\mathbf{N}')$, and $E_{\mathcal{P},j,f,\mathbf{v},\mathbf{w}}(\mathbf{N},\mathbf{N}')$ using the partial fraction decomposition from Proposition 1.3.2.

## 3.2 Value and derivative of $h_{\mathbf{N},\mathbf{N}',\mathcal{P},j,\mathbf{k}}(s)$ at $s=0$

Let $\mathcal{P} \subset [1, P]$, $j \in [1, Q]$, and $\mathbf{k} = (k_p)_{p \in \mathcal{P}} \in \mathbb{N}_0^{\mathcal{P}^c}$. In this section, we're now aiming to compute the coefficients $Q_{\mathbf{N},\mathbf{N}'}^0(\mathcal{P},j,\mathbf{k})$ and $Q_{\mathbf{N},\mathbf{N}'}^1(\mathcal{P},j,\mathbf{k})$, which we recall the definition:

$$ \begin{aligned} h_{\mathbf{N},\mathbf{N}',\mathcal{P},j,\mathbf{k}}(0) &= Q_{\mathbf{N},\mathbf{N}'}^0(\mathcal{P},j,\mathbf{k}), \\ \partial_s h_{\mathbf{N},\mathbf{N}',\mathcal{P},j,\mathbf{k}}(0) &= Q_{\mathbf{N},\mathbf{N}'}^1(\mathcal{P},j,\mathbf{k}). \end{aligned} $$

Let's recall that we prove in Proposition 2.2.2 that the function

$$ h_{\mathbf{N},\mathbf{N}',\mathcal{P},j,\mathbf{k}}(s) = \frac{1}{\prod_{q=1}^{Q} \Gamma(-N_q' + \mu_q' s)} \cdot \int_{0}^{1} \dots \int_{0}^{1} \prod_{\substack{q=1 \\ q \neq j}}^{Q} x_q^{-N_q' + \mu_q' s - 1} \prod_{p \in \mathcal{P}} l_p^*(\hat{x}^j)^{-N_p + \mu_p s - 1} \prod_{p \in \mathcal{P}^c} l_p^*(\hat{x}^j)^{k_p} dx_1 \dots \widehat{dx_j} \dots dx_Q $$

is holomorphic.

We also recall that we obtained from that proposition an expression of the derivative $h_{\mathbf{N},\mathbf{N}',\mathcal{P},j,\mathbf{k}}(s)$ at $s=0$.

### 3.2.1 Preliminaries

Let $\mathcal{P} \subset [1, P]$, $j \in [1, Q]$, and $0 < \epsilon \ll 1$. We know that for all $s \in \mathbb{C}$, we have

$$ h_{\mathbf{N},\mathbf{N}',\mathcal{P},j,\mathbf{k}}(s) = \sum_{Q \subset [1,Q]\setminus\{j\}} I_{\mathbf{N},\mathbf{N}',\mathcal{P},j,Q,\mathbf{k}}(\epsilon, s), $$
we then deduce the fact that it is enough to study the values and the derivative values of $I_{\mathbf{N},\mathbf{N}',\mathcal{P},j,\mathcal{Q},\mathbf{k}}(\epsilon, s)$ at $s = 0$ for $0 < \epsilon \ll 1$ in order to get the values and the derivative values of $h_{\mathbf{N},\mathbf{N}',\mathcal{P},j,\mathbf{k}}(s)$ at $s = 0$. We now recall point iii) obtained in Proposition 2.2.2:

Let $\mathbf{k} = (k_p)_{p \in \mathcal{P}^c} \in \mathbb{N}_0^{\mathcal{P}^c}$. For all $s \in \mathbb{C}$, and for $\epsilon > 0$ small enough, we have:

$$h_{\mathbf{N},\mathbf{N}',\mathcal{P},j,\mathbf{k}}(s) = \sum_{\mathcal{Q} \subset [1,\mathcal{Q}] \setminus \{j\}} \sum_{\mathbf{v}=(\mathbf{v}_p)_{p \in \mathcal{P}} \in (\mathbb{N}_0^{\mathcal{Q}})^\mathcal{P}} \sum_{\substack{\mathbf{w}=(\mathbf{w}_p)_{p \in \mathcal{P}^c} \in (\mathbb{N}_0^{\mathcal{Q}})^{\mathcal{P}^c} \\ \forall p \in \mathcal{P}^c | \mathbf{w}_p|=k_p}} \left( \prod_{p \in \mathcal{P}^c} \binom{k_p}{\mathbf{w}_p} \prod_{q \in [1,\mathcal{Q}]} c_{q,p}^{w_{p,q}} \right) \cdot \left( \prod_{p \in \mathcal{P}} \binom{-N_p + \mu_p s - 1}{|\mathbf{v}_p|} \binom{|\mathbf{v}_p|}{\mathbf{v}_p} \prod_{q \in \mathcal{Q}} c_{q,p}^{v_{p,q}} \right) \frac{1}{\prod_{q \in [1,\mathcal{Q}] \setminus \{j\}} \Gamma(-N'_q + \mu'_q s) \Gamma(-N'_q + \mu'_q s + \mathbf{v}(q) + \mathbf{w}(q))} \cdot \int_{\epsilon}^{1} \dots \int_{\epsilon}^{1} \prod_{q \in \mathcal{Q}^c} x_q^{-N'_q + \mu'_q s - 1 + \mathbf{w}(q)} \cdot \prod_{p \in \mathcal{P}} \left( c_{j,p} + \sum_{f \in \mathcal{Q}^c} c_{f,p} x_f \right)^{-N_p + \mu_p s - 1 - |\mathbf{v}_p|} \prod_{q \in \mathcal{Q}^c} dx_q.$$

Let's consider $T_{\mathbf{N},\mathbf{N}',\mathcal{P},j,\mathcal{Q},\mathbf{k},\mathbf{v},\mathbf{w}}(\epsilon, s)$ the general term of the above series:

$$T_{\mathbf{N},\mathbf{N}',\mathcal{P},j,\mathcal{Q},\mathbf{k},\mathbf{v},\mathbf{w}}(\epsilon, s) = \left( \prod_{p \in \mathcal{P}^c} \binom{k_p}{\mathbf{w}_p} \prod_{q=1}^{Q} c_{q,p}^{w_{p,q}} \right) \left( \prod_{p \in \mathcal{P}} \binom{-N_p + \mu_p s - 1}{|\mathbf{v}_p|} \binom{|\mathbf{v}_p|}{\mathbf{v}_p} \prod_{q \in \mathcal{Q}} c_{q,p}^{v_{p,q}} \right) \cdot \frac{1}{\prod_{q=1}^{Q} \Gamma(-N'_q + \mu'_q s + \mathbf{v}(q) + \mathbf{w}(q))} \cdot \frac{\epsilon^{\sum_{q \in \mathcal{Q}} (-N'_q + \mu'_q s + \mathbf{v}(q) + \mathbf{w}(q))}}{\prod_{q \neq j} \Gamma(-N'_q + \mu'_q s + \mathbf{v}(q) + \mathbf{w}(q))} \cdot \int_{\epsilon}^{1} \dots \int_{\epsilon}^{1} \prod_{q \in \mathcal{Q}^c} x_q^{-N'_q + \mu'_q s - 1 + \mathbf{w}(q)} \cdot \prod_{p \in \mathcal{P}} \left( c_{j,p} + \sum_{f \in \mathcal{Q}^c} c_{f,p} x_f \right)^{-N_p + \mu_p s - 1 - |\mathbf{v}_p|} \prod_{q \in \mathcal{Q}^c} dx_q. \quad (3.1)$$

Let's recall that, with that definition in mind, we have by point iii) of Lemma 2.3.22:

$$I_{\mathbf{N},\mathbf{N}',\mathcal{P},j,\mathcal{Q},\mathbf{k}}(\epsilon, 0) = \sum_{\substack{\mathbf{v}=(\mathbf{v}_p)_{p \in \mathcal{P}} \in (\mathbb{N}_0^{\mathcal{Q}})^\mathcal{P} \\ \forall p \in \mathcal{P}^c | \mathbf{w}_p|=k_p}} \sum_{\substack{\mathbf{w}=(\mathbf{w}_p)_{p \in \mathcal{P}^c} \in (\mathbb{N}_0^{\mathcal{Q}})^{\mathcal{P}^c} \\ \forall p \in \mathcal{P}^c | \mathbf{w}_p|=k_p}} T_{\mathbf{N},\mathbf{N}',\mathcal{P},j,\mathcal{Q},\mathbf{k},\mathbf{v},\mathbf{w}}(\epsilon, 0). \quad (3.2)$$

and by that same point iii), we also get an expression of the derivative at $s = 0$:

$$\left( \partial_s I_{\mathbf{N},\mathbf{N}',\mathcal{P},j,\mathcal{Q},\mathbf{k}}(\epsilon, s) \right)_{|s=0} = \sum_{\mathbf{v}=(\mathbf{v}_p)_{p \in \mathcal{P}} \in (\mathbb{N}_0^{\mathcal{Q}})^\mathcal{P}} \sum_{\substack{\mathbf{w}=(\mathbf{w}_p)_{p \in \mathcal{P}^c} \in (\mathbb{N}_0^{\mathcal{Q}})^{\mathcal{P}^c} \\ \forall p \in \mathcal{P}^c | \mathbf{w}_p|=k_p}} \left( \partial_s T_{\mathbf{N},\mathbf{N}',\mathcal{P},j,\mathcal{Q},\mathbf{k},\mathbf{v},\mathbf{w}}(\epsilon, s) \right)_{|s=0}. \quad (3.3)$$

Thanks to the formula (2.24), we get:

$$h_{\mathbf{N},\mathbf{N}',\mathcal{P},j,\mathbf{k}}(0) = \sum_{\mathcal{Q} \subset [1,\mathcal{Q}] \setminus \{j\}} I_{\mathbf{N},\mathbf{N}',\mathcal{P},j,\mathcal{Q},\mathbf{k}}(\epsilon, 0), \quad (3.4)$$

$$\left( \partial_s h_{\mathbf{N},\mathbf{N}',\mathcal{P},j,\mathbf{k}}(s) \right)_{|s=0} = \sum_{\mathcal{Q} \subset [1,\mathcal{Q}] \setminus \{j\}} \left( I_{\mathbf{N},\mathbf{N}',\mathcal{P},j,\mathcal{Q},\mathbf{k}}(\epsilon, s) \right)_{|s=0}. \quad (3.5)$$

Therefore, we can compute the coefficients $Q_{\mathbf{N},\mathbf{N}'}^0(\mathcal{P},j,\mathbf{k})$ and $Q_{\mathbf{N},\mathbf{N}'}^1(\mathcal{P},j,\mathbf{k})$ by computing the value and the derivative value of $T_{\mathbf{N},\mathbf{N}',\mathcal{P},j,\mathcal{Q},\mathbf{k},\mathbf{v},\mathbf{w}}(\epsilon, s)$ at $s = 0$.
**Lemma 3.2.1.**

$$ T_{\mathbf{N},\mathbf{N}',\mathcal{P},j,\mathcal{Q},\mathbf{k},\mathbf{v},\mathbf{w}}(\epsilon, s) = \begin{cases} O(1) & \text{if } |\mathcal{Q}| = Q-1, \text{ and } \forall q \in \mathcal{Q}, \qquad \mathbf{v}(q) + \mathbf{w}(q) = N'_q \\ O(s) & \text{if } |\mathcal{Q}| = Q-1 \text{ and } \exists f \in \mathcal{Q}\setminus\{j\}, \forall q \in \mathcal{Q}\setminus\{f\}, \qquad \mathbf{v}(q) + \mathbf{w}(q) = N'_q \\ O(s) & \text{if } |\mathcal{Q}| = Q-2 \text{ and } \forall q \in \mathcal{Q}, \qquad \mathbf{v}(f) + \mathbf{w}(f) \neq N'_f \\ O(s^2) & \text{else.} \end{cases} $$

*Proof.* We know that the meromorphic function

$$ s \mapsto \prod_{q \in \mathcal{Q}} \frac{1}{-N'_q + \mu'_q s + \mathbf{v}(q) + \mathbf{w}(q)} $$

has a pole of order

$$ |\{q \in \mathcal{Q} | N'_q = \mathbf{v}(q) + \mathbf{w}(q)\}| $$

at $s=0$. In particular, the order is necessarily smaller than $|\mathcal{Q}|$. By the property of the $\Gamma$ function, we know that the function $s \mapsto \frac{1}{\prod_{q=1}^Q \Gamma(-N'_q + s\mu'_q)}$ has a zero of order $Q-1$ at $s=0$. We then obtain that

$T_{\mathbf{N},\mathbf{N}',\mathcal{P},j,\mathcal{Q},\mathbf{k},\mathbf{v},\mathbf{w}}(\epsilon, s)$ has a zero of order

$$ n = Q - 1 - |\{q \in \mathcal{Q} | N'_q = \mathbf{v}(q) + \mathbf{w}(q)\}|. $$

□

### 3.2.2 Computation of $Q_{\mathbf{N},\mathbf{N}'}^0(\mathcal{P}, j, \mathbf{k})$

**Proposition 3.2.2.** Let $\mathcal{Q} \subset [1, \mathcal{Q}]\{j\}$.

* If $\mathcal{Q} = [1, \mathcal{Q}]\{j\}$ and $\forall q \in \mathcal{Q}$, $\mathbf{v}(q) + \mathbf{w}(q) = N'_q$, then

$$ T_{\mathbf{N},\mathbf{N}',\mathcal{P},j,\mathcal{Q},\mathbf{k},\mathbf{v},\mathbf{w}}(\epsilon, 0) = \left( \prod_{p \in \mathcal{P}^c} \binom{k_p}{\mathbf{w}_p} \prod_{q \in [1, \mathcal{Q}]} c_{q,p}^{w_{p,q}} \right) \left( \prod_{p \in \mathcal{P}} \binom{-N_p - 1}{|\mathbf{v}_p|} \binom{|\mathbf{v}_p|}{\mathbf{v}_p} \prod_{q \in \mathcal{Q}} c_{q,p}^{v_{p,q}} \right) \prod_{\substack{q=1 \\ q \neq j}}^Q (-1)^{N'_q} N'_q! $$

* Otherwise, we have

$$ T_{\mathbf{N},\mathbf{N}',\mathcal{P},j,\mathcal{Q},\mathbf{k},\mathbf{v},\mathbf{w}}(\epsilon, 0) = 0. $$

*Proof.* In the first case, by Lemma 3.1.1, we have that:

$$ \left( \frac{1}{\Gamma(-N'_q + \mu'_q s)} \frac{1}{\mu'_q s} \right)_{s=0} = (-1)^{N'_q} N'_q! \frac{\mu'_q}{\mu'_q}. $$

By Taking $s=0$ in the expression of $T_{\mathbf{N},\mathbf{N}',\mathcal{P},j,\mathcal{Q},\mathbf{k},\mathbf{v},\mathbf{w}}(\epsilon, s)$, we obtain the first formula from the proposition. The second formula follows from Lemma 3.2.1. □

By formula 3.2, and by the previous proposition, we get:

**Corollary 3.2.3.** Let $\mathcal{Q} \subset [1, \mathcal{Q}]\{j\}$, and $0 < \epsilon \ll 1$.
* If $\mathcal{Q} = [1, \mathcal{Q}] \setminus \{j\}$, then we have

$$ I_{\mathcal{N},\mathcal{N}',\mathcal{P},j,\mathcal{Q},\mathbf{k}}(\epsilon, 0) = (-1)^{|\mathcal{N}'|+|\{1,\mathcal{Q}\} \setminus \{j\}|} \left( \prod_{\substack{q=1 \\ q \neq j}}^{Q} N_q'! \right) \sum_{\substack{\mathbf{w}=(\mathbf{w}_p)_{p \in \mathcal{P}} \in (\mathbb{N}_0^Q)^{\mathcal{P}} \\ \forall p \in \mathcal{P}^c | \mathbf{w}_p| = k_p}} \sum_{\substack{\mathbf{v}=(\mathbf{v}_p)_{p \in \mathcal{P}} \in (\mathbb{N}_0^{[1,\mathcal{Q}] \setminus \{j\}})^{\mathcal{P}} \\ \forall q \neq j, \mathbf{v}(q)+\mathbf{w}(q)=N_q' }} \left( \prod_{p \in \mathcal{P}^c} \binom{k_p}{\mathbf{w}_p} \prod_{q=1}^{Q} c_{q,p}^{w_{p,q}} \right) \left( \prod_{p \in \mathcal{P}} \binom{-N_p-1}{|\mathbf{v}_p|} \binom{|\mathbf{v}_p|}{\mathbf{v}_p} \prod_{\substack{q=1 \\ q \neq j}}^{Q} c_{q,p}^{v_{p,q}} \right). $$

* Otherwise we have

$$ I_{\mathcal{N},\mathcal{N}',\mathcal{P},j,\mathcal{Q},\mathbf{k}}(\epsilon, 0) = 0. $$

By formula 2.24, we recall that we have

$$ h_{\mathcal{N},\mathcal{N}',\mathcal{P},j,\mathbf{k}}(0) = \sum_{\mathcal{Q} \subset [1,\mathcal{Q}] \setminus \{j\}} I_{\mathcal{N},\mathcal{N}',\mathcal{P},j,\mathcal{Q},\mathbf{k}}(\epsilon, 0), $$

by the previous corollary, we then obtain

$$ Q_{\mu,\mu'}^0(\mathcal{N},\mathcal{N}'(\mathcal{P},j,\mathcal{Q},\mathbf{k})) = (-1)^{|\mathcal{N}'|+|\{1,\mathcal{Q}\} \setminus \{j\}|} \left( \prod_{\substack{q=1 \\ q \neq j}}^{Q} N_q'! \right) \sum_{\substack{\mathbf{w}=(\mathbf{w}_p)_{p \in \mathcal{P}} \in (\mathbb{N}_0^Q)^{\mathcal{P}} \\ \forall p \in \mathcal{P}^c | \mathbf{w}_p| = k_p}} \sum_{\substack{\mathbf{v}=(\mathbf{v}_p)_{p \in \mathcal{P}} \in (\mathbb{N}_0^{[1,\mathcal{Q}] \setminus \{j\}})^{\mathcal{P}} \\ \forall q \neq j, \mathbf{v}(q)+\mathbf{w}(q)=N_q' }} \left( \prod_{p \in \mathcal{P}^c} \binom{k_p}{\mathbf{w}_p} \prod_{q=1}^{Q} c_{q,p}^{w_{p,q}} \right) \left( \prod_{p \in \mathcal{P}} \binom{-N_p-1}{|\mathbf{v}_p|} \binom{|\mathbf{v}_p|}{\mathbf{v}_p} \prod_{\substack{q=1 \\ q \neq j}}^{Q} c_{q,p}^{v_{p,q}} \right). \quad (3.6) $$

### 3.2.3 Computation of $Q_{\mu,\mu'}^1(\mathcal{N},\mathcal{N}'(\mathcal{P},j,\mathbf{k})$

Taking $s=0$ in the derivative with respect to $s$ of the general term

$$ T_{\mu,\mu'}^{\mathcal{N},\mathcal{N}',\mathcal{P},j,\mathcal{Q},\mathbf{k},\mathbf{v},\mathbf{w}}(\epsilon, s) = \left( \prod_{p \in \mathcal{P}^c} \binom{k_p}{\mathbf{w}_p} \prod_{q=1}^{Q} c_{q,p}^{w_{p,q}} \right) \left( \prod_{p \in \mathcal{P}} \binom{-N_p + \mu_p s - 1}{|\mathbf{v}_p|} \binom{|\mathbf{v}_p|}{\mathbf{v}_p} \prod_{q \in \mathcal{Q}} c_{q,p}^{v_{p,q}} \right) \\ \cdot \frac{1}{\prod_{q=1}^{Q} \Gamma(-N_q' + \mu_q' s) \prod_{q \in \mathcal{Q}} (-N_q' + \mu_q' s + \mathbf{v}(q) + \mathbf{w}(q))} \\ \cdot \int_{\epsilon}^{1} \dots \int_{\epsilon}^{1} \prod_{q \in \mathcal{Q}^c} x_q^{-N_q' + \mu_q' s - 1 + \mathbf{w}(q)} \cdot \prod_{p \in \mathcal{P}} \left( c_{j,p} + \sum_{f \in \mathcal{Q}^c} c_{f,p} x_f \right)^{-N_p + \mu_p s - 1 - |\mathbf{v}_p|} \prod_{q \in \mathcal{Q}^c} dx_q, $$

we get:

**Proposition 3.2.4.** Let $\mathcal{Q} \subset \mathcal{Q} = [1, \mathcal{Q}] \setminus \{j\}$,
* If $\mathcal{Q} = [1, \mathcal{Q}] \setminus \{j\}$ and $\forall q \in \mathcal{Q}$, $\mathbf{v}(q) + \mathbf{w}(q) = N'_q$, we have

$$ \partial_s \left( T_{\mathbf{N},\mathbf{N}',\mathcal{P},j,\mathcal{Q},\mathbf{k},\mathbf{v},\mathbf{w}}(\epsilon, s) \right) \Big|_{s=0} = \qquad (3.7) $$

$$ \begin{aligned} & (-1)^{|\mathbf{N}'|} |[1, \mathcal{Q}] \setminus \{j\}| \left( \prod_{p \in \mathcal{P}^c} \binom{k_p}{\mathbf{w}_p} \prod_{q=1}^Q c_{q,p}^{w_{p,q}} \right) \left( \prod_{p \in \mathcal{P}} \binom{-N_p - 1}{|\mathbf{v}_p|} \binom{|\mathbf{v}_p|}{\mathbf{v}_p} \prod_{\substack{q=1 \\ q \neq j}}^Q c_{q,p}^{v_{p,q}} \right) \prod_{\substack{q=1 \\ q \neq j}}^Q N'_q! \\ & \qquad \cdot \left( \sum_{p \in \mathcal{P}} \mu_p (h_{N_p} - h_{N_p + |\mathbf{v}_p|}) + \sum_{\substack{q=1 \\ q \neq j}}^Q \mu'_q (\gamma - h_{N'_q}) \right) \\ & \qquad + (-1)^{|\mathbf{N}'|} |[1, \mathcal{Q}] \setminus \{j\}| |\boldsymbol{\mu}'|_{\mathcal{Q}} \left( \prod_{p \in \mathcal{P}^c} \binom{k_p}{\mathbf{w}_p} \prod_{q=1}^Q c_{q,p}^{w_{p,q}} \right) \left( \prod_{p \in \mathcal{P}} \binom{-N_p - 1}{|\mathbf{v}_p|} \binom{|\mathbf{v}_p|}{\mathbf{v}_p} \prod_{\substack{q=1 \\ q \neq j}}^Q c_{q,p}^{v_{p,q}} \right) \left( \prod_{\substack{q=1 \\ q \neq j}}^Q N'_q! \right) \ln(\epsilon). \end{aligned} $$

* If $\mathcal{Q} = [1, \mathcal{Q}] \setminus \{j\}$ and if there exist an integer $f \in \mathcal{Q} \setminus \{j\}$ such that $\mathbf{v}(f) + \mathbf{w}(f) \neq N'_f$ and such that for all $q \in \mathcal{Q} \setminus \{f\}$, $\mathbf{v}(q) + \mathbf{w}(q) = N'_q$, we then have

$$ \partial_s \left( T_{\mathbf{N},\mathbf{N}',\mathcal{P},j,\mathcal{Q},\mathbf{k},\mathbf{v},\mathbf{w}}(\epsilon, s) \right) \Big|_{s=0} = \qquad (3.8) $$

$$ \begin{aligned} & (-1)^{|\mathbf{N}'|} |[1, \mathcal{Q}] \setminus \{j\}| \left( \prod_{\substack{q=1 \\ q \neq j}}^Q N'_q! \right) \left( \prod_{p \in \mathcal{P}^c} \binom{k_p}{\mathbf{w}_p} \prod_{q=1}^Q c_{q,p}^{w_{p,q}} \right) \left( \prod_{p \in \mathcal{P}} \binom{-N_p - 1}{|\mathbf{v}_p|} \binom{|\mathbf{v}_p|}{\mathbf{v}_p} \prod_{\substack{q=1 \\ q \neq j}}^Q c_{q,p}^{v_{p,q}} \right) \\ & \qquad \cdot \frac{\mu'_f}{-N'_f + \mathbf{v}(f) + \mathbf{w}(f)} \epsilon^{-N'_f + \mathbf{v}(f) + \mathbf{w}(f)}. \end{aligned} $$

* If $\mathcal{Q} = [1, \mathcal{Q}] \setminus \{j, f\}$ such that $\mathbf{v}(q) + \mathbf{w}(q) = N'_q$ for all $q \in \mathcal{Q}$, then there exist a function $\beta_{\mathcal{P},j,f,\mathbf{v},\mathbf{w},\mathbf{N},\mathbf{N}'}^*(\epsilon)$ which has a decomposition as a sum between a multiple of $\ln(\epsilon)$, and a Laurent series in $\epsilon$, for $0 < \epsilon$ small enough, without any constant term and such that

$$ \partial_s \left( T_{\mathbf{N},\mathbf{N}',\mathcal{P},j,\mathcal{Q},\mathbf{k},\mathbf{v},\mathbf{w}}(\epsilon, s) \right) \Big|_{s=0} = \qquad (3.9) $$

$$ \mu'_f(-1)^{|\mathbf{N}'|} |[1, \mathcal{Q}] \setminus \{j\}| \left( \prod_{\substack{q=1 \\ q \neq j}}^{Q} N'_q! \right) \left( \prod_{p \in \mathcal{P}^c} \binom{k_p}{\mathbf{w}_p} \prod_{q=1}^{Q} c_{q,p}^{w_{p,q}} \right) \left( \prod_{p \in \mathcal{P}} \binom{-N_p - 1}{|\mathbf{v}_p|} \binom{|\mathbf{v}_p|}{\mathbf{v}_p} \prod_{\substack{q=1 \\ q \neq j, f}}^{Q} c_{q,p}^{v_{p,q}} \right) F_{\mathcal{P},j,f,\mathbf{v},\mathbf{w}}(\mathbf{N}, \mathbf{N}') \\ + \beta_{\mathcal{P},j,f,\mathbf{v},\mathbf{w},\mathbf{N},\mathbf{N}'}^*(\epsilon), $$

with $F_{\mathcal{P},j,f,\mathbf{v},\mathbf{w}}(\mathbf{N}, \mathbf{N}')$ being the constant defined in Proposition 1.3.2.

* Otherwise we have

$$ \partial_s \left( T_{\mathbf{N},\mathbf{N}',\mathcal{P},j,\mathcal{Q},\mathbf{k},\mathbf{v},\mathbf{w}}(\epsilon, s) \right) \Big|_{s=0} = 0. $$
*Proof.* • If $\mathcal{Q} = [1, \mathcal{Q}] \setminus \{j\}$, we then find that

$$
\begin{align*}
\partial_s \left( T_{\mathcal{N},\mathcal{N}',\mathcal{P},j,\mathcal{Q},\mathbf{k},\mathbf{v},\mathbf{w}}(\epsilon, s) \right)_{|s=0} &= \\
& \quad \left( \prod_{p \in \mathcal{P}^c} \binom{k_p}{\mathbf{w}_p} \prod_{q=1}^Q c_{q,p}^{w_{p,q}} \right) \partial_s \left( \prod_{p \in \mathcal{P}} \binom{-N_p + \mu_p s - 1}{|\mathbf{v}_p|} \binom{|\mathbf{v}_p|}{\mathbf{v}_p} \prod_{\substack{q=1 \\ q \neq j}}^Q c_{q,p}^{v_{p,q}} \right)_{|s=0} \\
& \quad \cdot \left( \frac{1}{\prod_{q=1}^Q \Gamma(-N'_q + \mu'_q s)(-N'_q + \mu'_q s + \mathbf{v}(q) + \mathbf{w}(q))} \right)_{|s=0} \epsilon^{\sum_{q=1}^Q (-N'_q + \mathbf{v}(q) + \mathbf{w}(q))} \\
& \quad + \left( \prod_{p \in \mathcal{P}^c} \binom{k_p}{\mathbf{w}_p} \prod_{q=1}^Q c_{q,p}^{w_{p,q}} \right) \left( \prod_{p \in \mathcal{P}} \binom{-N_p - 1}{|\mathbf{v}_p|} \binom{|\mathbf{v}_p|}{\mathbf{v}_p} \prod_{\substack{q=1 \\ q \neq j}}^Q c_{q,p}^{v_{p,q}} \right) \\
& \quad \cdot \partial_s \left( \frac{1}{\prod_{q=1}^Q \Gamma(-N'_q + \mu'_q s)(-N'_q + \mu'_q s + \mathbf{v}(q) + \mathbf{w}(q))} \right)_{|s=0} \epsilon^{\sum_{q=1}^Q (-N'_q + \mathbf{v}(q) + \mathbf{w}(q))} \\
& \quad + |\boldsymbol{\mu}'|_{\mathcal{Q}} \left( \prod_{p \in \mathcal{P}^c} \binom{k_p}{\mathbf{w}_p} \prod_{q=1}^Q c_{q,p}^{w_{p,q}} \right) \left( \prod_{p \in \mathcal{P}} \binom{-N_p - 1}{|\mathbf{v}_p|} \binom{|\mathbf{v}_p|}{\mathbf{v}_p} \prod_{\substack{q=1 \\ q \neq j}}^Q c_{q,p}^{v_{p,q}} \right) \\
& \quad \cdot \left( \frac{1}{\prod_{q=1}^Q \Gamma(-N'_q + \mu'_q s)(-N'_q + \mu'_q s + \mathbf{v}(q) + \mathbf{w}(q))} \right)_{|s=0} \ln(\epsilon) \epsilon^{\sum_{q=1}^Q (-N'_q + \mathbf{v}(q) + \mathbf{w}(q))}
\end{align*}
$$

By Lemma 3.1.2, we get

$$
\partial_s \left( \prod_{p \in \mathcal{P}} \binom{-N_p + \mu_p s - 1}{|\mathbf{v}_p|} \right)_{|s=0} = \left( \sum_{p \in \mathcal{P}} \mu_p (h_{N_p} - h_{N_p + |\mathbf{v}_p|}) \right) \prod_{p \in \mathcal{P}} \binom{-N_p - 1}{|\mathbf{v}_p|},
$$

which concludes the first formula.

• If $\mathcal{Q} = [1, \mathcal{Q}] \setminus \{j\}$ and $\forall q \in \mathcal{Q}$, $\mathbf{v}(q) + \mathbf{w}(q) = N'_q$, we then get by Lemma 3.1.1,

$$
\begin{align*}
\partial_s \left( \frac{1}{\prod_{q \neq j}^{Q} \Gamma(-N'_q + \mu'_q s) \mu'_q s} \right)_{|s=0} &= \sum_{\substack{f=1 \\ f \neq j}}^{Q} \mu'_f (-1)^{N'_f} N'_f! (\gamma - h_{N'_f}) \prod_{\substack{q=1 \\ q \neq j, f}}^{Q} (-1)^{N'_q} N'_q! \\
&= (-1)^{|\mathbf{N}'|} \mathbb{1}_{[1, \mathcal{Q}] \setminus \{j\}} \left( \prod_{\substack{q=1 \\ q \neq j}}^{Q} N'_q! \right) \left( \sum_{\substack{q=1 \\ q \neq j}}^{Q} \mu'_q (\gamma - h_{N'_q}) \right) \\
\left( \frac{1}{\prod_{q \neq j}^{Q} \Gamma(-N'_q + \mu'_q s) \mu'_q s} \right)_{|s=0} &= (-1)^{|\mathbf{N}'|} \mathbb{1}_{[1, \mathcal{Q}] \setminus \{j\}} \prod_{\substack{q=1 \\ q \neq j}}^{Q} N'_q!.
\end{align*}
$$
Therefore we find that

$$
\begin{aligned}
& \partial_s \left( T_{\mathbf{N},\mathbf{N}',\mathcal{P},j,\mathbf{k},\mathbf{v},\mathbf{w}}(\epsilon, s) \right) \Big|_{s=0} \\
&= \left( \prod_{p \in \mathcal{P}^c} \left( \frac{k_p}{\mathbf{w}_p} \right) \prod_{q=1}^Q c_{q,p}^{w_{p,q}} \right) \left( \sum_{p \in \mathcal{P}} \mu_p (h_{N_p} - h_{N_p+|\mathbf{v}_p|}) \right) \left( \prod_{\substack{p \in \mathcal{P} \\ q=1}} \left( \frac{-N_p-1}{|\mathbf{v}_p|} \right) \left( \frac{|\mathbf{v}_p|}{|\mathbf{v}_p|} \right) \prod_{\substack{q=1 \\ q \neq j}}^Q c_{q,p}^{v_{p,q}} \right) \\
& \quad \cdot (-1)^{|\mathbf{N}'||[1,Q]\setminus\{j\}|} \prod_{\substack{q=1 \\ q \neq j}}^Q N'_q! \\
& \quad + \left( \prod_{p \in \mathcal{P}^c} \left( \frac{k_p}{\mathbf{w}_p} \right) \prod_{q=1}^Q c_{q,p}^{w_{p,q}} \right) \left( \prod_{\substack{p \in \mathcal{P} \\ q=1}} \left( \frac{-N_p-1}{|\mathbf{v}_p|} \right) \left( \frac{|\mathbf{v}_p|}{|\mathbf{v}_p|} \right) \prod_{\substack{q=1 \\ q \neq j}}^Q c_{q,p}^{v_{p,q}} \right) \\
& \quad \cdot (-1)^{|\mathbf{N}'||[1,Q]\setminus\{j\}|} \left( \prod_{\substack{q=1 \\ q \neq j}}^Q N'_q! \right) \left( \sum_{\substack{q=1 \\ q \neq j}}^Q \mu'_q (\gamma - h_{N'_q}) \right) \\
& \quad + |\boldsymbol{\mu}'|_{\mathcal{Q}} \left( \prod_{p \in \mathcal{P}^c} \left( \frac{k_p}{\mathbf{w}_p} \right) \prod_{q=1}^Q c_{q,p}^{w_{p,q}} \right) \left( \prod_{\substack{p \in \mathcal{P} \\ q=1}} \left( \frac{-N_p-1}{|\mathbf{v}_p|} \right) \left( \frac{|\mathbf{v}_p|}{|\mathbf{v}_p|} \right) \prod_{\substack{q=1 \\ q \neq j}}^Q c_{q,p}^{v_{p,q}} \right) \\
& \quad \cdot (-1)^{|\mathbf{N}'||[1,Q]\setminus\{j\}|} \prod_{\substack{q=1 \\ q \neq j}}^Q N'_q! \ln(\epsilon).
\end{aligned}
$$

Factoring the first and second term of this last sum, we get 3.7.

• If $\mathcal{Q} = [1, \mathcal{Q}] \setminus \{j\}$ and if there exist an integer $f \in \mathcal{Q} \setminus \{j\}$ such that $\mathbf{v}(f) + \mathbf{w}(f) \neq N'_f$ and such that for all $q \in \mathcal{Q} \setminus \{f\}$, we have $\mathbf{v}(q) + \mathbf{w}(q) = N'_q$, we then find by Lemma 3.1.1,

$$
\begin{aligned}
& \left( \frac{1}{\Gamma(-N'_f + \mu'_f s)(-N'_f + \mu'_f s + \mathbf{v}(f) + \mathbf{w}(f)) \prod_{q=1}^Q \Gamma(-N'_q + \mu'_q s) \mu'_q s} \right) \Big|_{s=0} = 0, \\
& \partial_s \left( \frac{1}{\Gamma(-N'_f + \mu'_f s)(-N'_f + \mu'_f s + \mathbf{v}(f) + \mathbf{w}(f)) \prod_{q=1}^Q \Gamma(-N'_q + \mu'_q s) \mu'_q s} \right) \Big|_{s=0} = \\
& (-1)^{|\mathbf{N}'|[1,Q]\setminus\{j\}} \left( \prod_{\substack{q=1 \\ q \neq j}}^Q N'_q! \right) \frac{\mu'_f}{-N'_f + \mathbf{v}(f) + \mathbf{w}(f)}.
\end{aligned}
$$
Therefore, we get

$$
\begin{align*}
\partial_s \left( T_{\mathbf{N},\mathbf{N}',\mathcal{P},j,\mathcal{Q},\mathbf{k},\mathbf{v},\mathbf{w}}(\epsilon, s) \right) \Big|_{s=0} &= \\
& \quad \left( \prod_{p \in \mathcal{P}^c} \left( \frac{k_p}{\mathbf{w}_p} \right) \prod_{q=1}^Q c_{q,p}^{w_p,q} \right) \left( \prod_{p \in \mathcal{P}} \left( \frac{-N_p - 1}{|\mathbf{v}_p|} \right) \left( \frac{|\mathbf{v}_p|}{|\mathbf{v}_p|} \right) \prod_{\substack{q=1 \\ q \neq j}}^Q c_{q,p}^{v_p,q} \right) \\
& \quad \cdot \partial_s \left( \frac{1}{\Gamma(-N_f' + \mu_f's)(-N_f' + \mu_f's + \mathbf{v}(f) + \mathbf{w}(f)) \prod_{q \neq j,f}^Q \Gamma(-N_q' + \mu_q's)(-N_q' + \mu_q's + \mathbf{v}(q) + \mathbf{w}(q))} \right) \Big|_{s=0} \\
& \quad \cdot \frac{\sum_{q=1}^Q (-N_q' + \mathbf{v}(q) + \mathbf{w}(q))}{\epsilon \sum_{\substack{q=1 \\ q \neq j}}^Q (-N_q' + \mathbf{v}(q) + \mathbf{w}(q))} .
\end{align*}
$$

Noticing that $\sum_{q=1}^{Q} (-N'_{q} + \mathbf{v}(q) + \mathbf{w}(q)) = -N'_{f} + \mathbf{v}(f) + \mathbf{w}(f) \neq 0$, we then obtain formula 3.8.

• If $\mathcal{Q} = [1, \mathcal{Q}] \setminus \{j, f\}$ such that $\mathbf{v}(q) + \mathbf{w}(q) = N'_{q}$ for all $q \in \mathcal{Q}$, we then find by Lemma 3.1.1 that

$$
\begin{gather*}
\left( \frac{1}{\Gamma(-N'_{f} + \mu'_{f}s) \prod_{q \neq j, f}^{Q} \Gamma(-N'_{q} + \mu'_{q}s) \mu'_{q}s} \right) \Big|_{s=0} = 0 \\
\partial_{s} \left( \frac{1}{\Gamma(-N'_{f} + \mu'_{f}s) \prod_{q \neq j, f}^{Q} \Gamma(-N'_{q} + \mu'_{q}s) \mu'_{q}s} \right) \Big|_{s=0} = \mu'_{f} \prod_{\substack{q=1 \\ q \neq j}}^{Q} (-1)^{N'_{q}} N'_{q} ! .
\end{gather*}
$$

Deriving the general term $T_{\mathbf{N},\mathbf{N}',\mathcal{P},j,\mathcal{Q},\mathbf{k},\mathbf{v},\mathbf{w}}(\epsilon, s)$ with respect to $s$, and by taking $s = 0$ we then find that

$$
\begin{align}
\partial_s \left( T_{\mu, \mu', \mathcal{P}, j, \mathcal{Q}, \mathbf{k}, \mathbf{v}, \mathbf{w}} (\epsilon, s) \right) \Big|_{s=0} &= \left( \prod_{p \in \mathcal{P}^c} \left( \frac{k_p}{\mathbf{w}_p} \right) \prod_{q=1}^Q c_{q,p}^{w_p,q} \right) \left( \prod_{p \in \mathcal{P}} \left( \frac{-N_p - 1}{|\mathbf{v}_p|} \right) \left( \frac{|\mathbf{v}_p|}{|\mathbf{v}_p|} \right) \prod_{\substack{q=1 \\ q \neq j, f}}^Q c_{q,p}^{v_p,q} \right) \tag{3.10} \\
&\quad \cdot \partial_s \left( \frac{1}{\Gamma(-N_f' + \mu_f's) \prod_{q=1}^Q \Gamma(-N_q' + \mu_q's) \mu_q's} \right) \Big|_{s=0} \quad \sum_{\substack{q=1 \\ q \neq j, f}}^Q (-N_q' + \mathbf{v}(q) + \mathbf{w}(q)) \nonumber \\
&\quad \cdot \int_{\epsilon}^{1} x_f^{-N_f' - 1 + \mathbf{w}(f)} \cdot \prod_{p \in \mathcal{P}} (c_{j,p} + c_{f,p}x_f)^{-N_p - 1 - |\mathbf{v}_p|} dx_f. \nonumber
\end{align}
$$

With the conditions on the index **v** and on the index **w**, we find that $\sum_{\substack{q=1 \\ q \neq j, f}}^{Q} (-N'_{q} + \mathbf{v}(q) + \mathbf{w}(q)) = 0$. By Lemma 3.1.4, we get

$$
\int_{\epsilon}^{1} x_{f}^{-N'_{f}-1+\mathbf{w}(f)} \cdot \prod_{p \in \mathcal{P}} (c_{j,p}+c_{f,p}x_{f})^{-N_{p}-1-|\mathbf{v}_{p}|} dx_{f} = \alpha_{\mathcal{P},j,f,\mathbf{v},\mathbf{w},\mathbf{N},\mathbf{N}'}^{*}(\epsilon) + F_{\mathcal{P},j,f,\mathbf{v},\mathbf{w}}(\mathbf{N},\mathbf{N}'),
$$

with $F_{\mathcal{P},j,f,\mathbf{v},\mathbf{w}}(\mathbf{N},\mathbf{N}')$ defined in Proposition 1.3.2, and $\alpha_{\mathcal{P},j,f,\mathbf{v},\mathbf{w},\mathbf{N},\mathbf{N}'}^*(\epsilon)$ being a function in $\epsilon$ of the form $a \ln(\epsilon) + \sum_{n \ge m} b_n \epsilon^n, m \in \mathbb{Z}$ and $a, b_n$ being complex numbers with $b_0 = 0$.
Injecting those last expressions into formula 3.10, we then get

$$
\begin{aligned}
\partial_s \left( T_{\mathbf{N},\mathbf{N}',\mathcal{P},j,\mathcal{Q},\mathbf{k},\mathbf{v},\mathbf{w}}(\epsilon, s) \right) \Big|_{s=0} = & \\
& \mu_f' \prod_{\substack{q=1 \\ q \neq j}} (-1)^{N'_q} N'_q! \left( \prod_{p \in \mathcal{P}^c} \binom{k_p}{\mathbf{w}_p} \prod_{q=1}^Q c_{q,p}^{w_{p,q}} \right) \left( \prod_{p \in \mathcal{P}} \binom{-N_p - 1}{|\mathbf{v}_p|} \binom{|\mathbf{v}_p|}{\mathbf{v}_p} \prod_{\substack{q=1 \\ q \neq j, f}}^Q c_{q,p}^{v_{p,q}} \right) \\
& \cdot \left( \alpha_{\mathcal{P},j,f,\mathbf{v},\mathbf{w},\mathbf{N},\mathbf{N}'}^*(\epsilon) + F_{\mathcal{P},j,f,\mathbf{v},\mathbf{w}}(\mathbf{N}, \mathbf{N}') \right).
\end{aligned}
$$

Let's define

$$
\beta_{\mathcal{P},j,f,\mathbf{v},\mathbf{w},\mathbf{N},\mathbf{N}'}^*(\epsilon) = \mu_f' \prod_{\substack{q=1 \\ q \neq j}} (-1)^{N'_q} N'_q! \left( \prod_{p \in \mathcal{P}^c} \binom{k_p}{\mathbf{w}_p} \prod_{q=1}^Q c_{q,p}^{w_{p,q}} \right) \left( \prod_{p \in \mathcal{P}} \binom{-N_p - 1}{|\mathbf{v}_p|} \binom{|\mathbf{v}_p|}{\mathbf{v}_p} \prod_{\substack{q=1 \\ q \neq j, f}}^Q c_{q,p}^{v_{p,q}} \right) \alpha_{\mathcal{P},j,f,\mathbf{v},\mathbf{w},\mathbf{N},\mathbf{N}'}^*(\epsilon)
$$

we then get the formula 3.9.

• If we're not in one of these three last cases above, therefore by Lemma 3.2.1, we find that

$$
T_{\mathbf{N},\mathbf{N}',\mathcal{P},j,\mathcal{Q},\mathbf{k},\mathbf{v},\mathbf{w}}(\epsilon, s) = O(s^2),
$$

which concludes this particular point.

<figure><img src="image_18.png" /></figure>

By formula 3.3 and by the previous proposition, we then find the following corollary:

**Corollary 3.2.5.** Let $\mathcal{Q} \subset [1, \mathcal{Q}] \setminus \{j\}$.

• If $\mathcal{Q} = [1, \mathcal{Q}] \setminus \{j\}$, then there exist a function $I_{\mathbf{N},\mathbf{N}',\mathcal{P},j,\mathcal{Q},\mathbf{k}}^*(\epsilon)$ which corresponds to a sum of a multiple of $\ln(\epsilon)$ and a Laurent series defined for all $0 < \epsilon \ll 1$, without any constant term, and such that

$$
\begin{aligned}
\partial_s \left( I_{\mathbf{N},\mathbf{N}',\mathcal{P},j,\llbracket \mathcal{Q} \rrbracket \setminus \{j\},\mathbf{k}}(\epsilon, s) \right) \Big|_{s=0} = & \\
& (-1)^{|\mathbf{N}'|} \prod_{\substack{q=1 \\ q \neq j}}^Q N'_q! \sum_{\mathbf{w}=(\mathbf{w}_p)_{p \in \mathcal{P}^c} \in (\mathbb{N}_0^{\mathcal{Q}})^{\mathcal{P}^c} \atop \forall p \in \mathcal{P}^c | \mathbf{w}_p| = k_p} \sum_{\mathbf{v}=(\mathbf{v}_p)_{p \in \mathcal{P}} \in (\mathbb{N}_0^{\llbracket 1, \mathcal{Q} \rrbracket} \setminus \{j\})^{\mathcal{P}} \atop \forall q \neq j, \mathbf{v}(q) + \mathbf{w}(q) = N'_q} \left( \prod_{p \in \mathcal{P}} \binom{k_p}{\mathbf{w}_p} \prod_{q=1}^Q c_{q,p}^{w_{p,q}} \right) \\
& \left( \prod_{p \in \mathcal{P}} \binom{-N_p - 1}{|\mathbf{v}_p|} \binom{|\mathbf{v}_p|}{\mathbf{v}_p} \prod_{\substack{q=1 \\ q \neq j}}^Q c_{q,p}^{v_{p,q}} \right) \left( \sum_{p \in \mathcal{P}} \mu_p(h_{N_p} - h_{N_p + |\mathbf{v}_p|}) + \sum_{\substack{q=1 \\ q \neq j}}^Q \mu'_q(\gamma - h_{N'_q}) \right) \\
& + I_{\mathbf{N},\mathbf{N}',\mathcal{P},j,\mathcal{Q},\mathbf{k}}^*(\epsilon).
\end{aligned}
\tag{3.11}
$$

• If $\mathcal{Q} = [1, \mathcal{Q}] \setminus \{j, f\}$, then we find that there is a function $I_{\mathbf{N},\mathbf{N}',\mathcal{P},j,\mathcal{Q},\mathbf{k}}^*(\epsilon)$ which corresponds to a sum of a multiple of $\ln(\epsilon)$ and a Laurent series defined for all $0 < \epsilon \ll 1$ without any constant term,
and such that

$$ \partial_s \left( I_{\mu, \mu'}^{\mathcal{N}, \mathcal{N}', \mathcal{P}, j, \mathcal{Q}, \mathbf{k}}(\epsilon, s) \right) \Big|_{s=0} = \qquad (3.12) $$

$$ I_{\mu, \mu'}^{\mathcal{N}, \mathcal{N}', \mathcal{P}, j, \mathcal{Q}, \mathbf{k}}(\epsilon) + (-1)^{|\mathcal{N}'|\llbracket 1, Q \rrbracket \setminus \{j\}} \mu_f' \left( \prod_{\substack{q=1 \\ q \neq j}}^{Q} N_q'! \right) \sum_{\substack{\mathbf{w}=(\mathbf{w}_p)_{p \in \mathcal{P}} \in (\mathbb{N}_0^Q)^{\mathcal{P}} \\ \forall p \in \mathcal{P}^c |\mathbf{w}_p|=k_p}} \sum_{\substack{\mathbf{v}=(\mathbf{v}_p)_{p \in \mathcal{P}} \in (\mathbb{N}_0^{[1,Q] \setminus \{j,f\}})^{\mathcal{P}} \\ \forall q \neq j, f, \mathbf{v}(q)+\mathbf{w}(q)=N_q'}} \left( \prod_{p \in \mathcal{P}^c} \binom{k_p}{\mathbf{w}_p} \prod_{q=1}^{Q} c_{q,p}^{w_{p,q}} \right) \left( \prod_{p \in \mathcal{P}} \binom{-N_p-1}{|\mathbf{v}_p|} \left( \frac{|\mathbf{v}_p|}{|\mathbf{v}_p|} \right)^Q \prod_{\substack{q=1 \\ q \neq j, f}}^{Q} c_{q,p}^{v_{p,q}} \right) F_{\mathcal{P}, j, f, \mathbf{v}, \mathbf{w}}(\mathbf{N}, \mathcal{N}') $$

with $F_{\mathcal{P},j,f,\mathbf{v},\mathbf{w}}(\mathbf{N}, \mathcal{N}')$ defined in Proposition 1.3.2.

• Otherwise, we find that

$$ \partial_s \left( T_{\mu, \mu'}^{\mathcal{N}, \mathcal{N}', \mathcal{P}, j, \mathcal{Q}, \mathbf{k}, \mathbf{v}, \mathbf{w}}(\epsilon, s) \right) \Big|_{s=0} = 0. $$

By formula 3.5, we then get that

$$ \partial_s \left( h_{\mu, \mu'}^{\mathcal{N}, \mathcal{N}', \mathcal{P}, j, \mathbf{k}}(s) \right) \Big|_{s=0} = \sum_{Q \subset [1, Q] \setminus \{j\}} \partial_s \left( I_{\mu, \mu'}^{\mathcal{N}, \mathcal{N}', \mathcal{P}, j, \mathcal{Q}, \mathbf{k}}(\epsilon, s) \right) \Big|_{s=0}, $$

therefore by the previous corollary we obtain the following expression for the coefficient $Q_{\mu,\mu'}^{1,N,N'}(\mathcal{P},j,\mathbf{k})$:

$$ \begin{aligned} Q_{\mu, \mu'}^{1, \mathcal{N}, \mathcal{N}'}(\mathcal{P}, j, \mathbf{k}) = & \partial_s \left( I_{\mu, \mu'}^{\mathcal{N}, \mathcal{N}', \mathcal{P}, j, \llbracket 1, Q \rrbracket \setminus \{j\}, \mathbf{k}}(\epsilon, s) \right) \Big|_{s=0} + \sum_{\substack{f=1 \\ f \neq j}}^{Q} \partial_s \left( I_{\mu, \mu'}^{\mathcal{N}, \mathcal{N}', \mathcal{P}, j, \llbracket 1, Q \rrbracket \setminus \{j, f\}, \mathbf{k}}(\epsilon, s) \right) \Big|_{s=0} \\ = & (-1)^{|\mathcal{N}'|\llbracket 1, Q \rrbracket \setminus \{j\}} \prod_{\substack{q=1 \\ q \neq j}}^{Q} N_q'! \sum_{\substack{\mathbf{w}=(\mathbf{w}_p)_{p \in \mathcal{P}} \in (\mathbb{N}_0^Q)^{\mathcal{P}} \\ \forall p \in \mathcal{P}^c |\mathbf{w}_p|=k_p}} \sum_{\substack{\mathbf{v}=(\mathbf{v}_p)_{p \in \mathcal{P}} \in (\mathbb{N}_0^{[1,Q] \setminus \{j\}})^{\mathcal{P}} \\ \forall q \neq j, \mathbf{v}(q)+\mathbf{w}(q)=N_q'}} \left( \prod_{p \in \mathcal{P}^c} \binom{k_p}{\mathbf{w}_p} \prod_{q=1}^{Q} c_{q,p}^{w_{p,q}} \right) \\ & \left( \prod_{p \in \mathcal{P}} \binom{-N_p-1}{|\mathbf{v}_p|} \left( \frac{|\mathbf{v}_p|}{|\mathbf{v}_p|} \right)^Q \prod_{\substack{q=1 \\ q \neq j}}^{Q} c_{q,p}^{v_{p,q}} \right) \left( \sum_{p \in \mathcal{P}} \mu_p (h_{N_p} - h_{N_p+|\mathbf{v}_p|}) + \sum_{\substack{q=1 \\ q \neq j}}^{Q} \mu_q' (\gamma - h_{N_q'}) \right) \\ + & (-1)^{|\mathcal{N}'|\llbracket 1, Q \rrbracket \setminus \{j\}} \left( \prod_{\substack{q=1 \\ q \neq j}}^{Q} N_q'! \right) \sum_{\substack{f=1 \\ f \neq j}}^{Q} \mu_f' \sum_{\substack{\mathbf{w}=(\mathbf{w}_p)_{p \in \mathcal{P}} \in (\mathbb{N}_0^Q)^{\mathcal{P}} \\ \forall p \in \mathcal{P}^c |\mathbf{w}_p|=k_p}} \sum_{\substack{\mathbf{v}=(\mathbf{v}_p)_{p \in \mathcal{P}} \in (\mathbb{N}_0^{[1,Q] \setminus \{j,f\}})^{\mathcal{P}} \\ \forall q \neq j, \mathbf{v}(q)+\mathbf{w}(q)=N_q'}} \left( \prod_{p \in \mathcal{P}^c} \binom{k_p}{\mathbf{w}_p} \prod_{q=1}^{Q} c_{q,p}^{w_{p,q}} \right) F_{\mathcal{P}, j, f, \mathbf{v}, \mathbf{w}}(\mathbf{N}, \mathcal{N}') \\ & + I_{\mu, \mu'}^{\mathcal{N}, \mathcal{N}', \mathcal{P}, j, \llbracket 1, Q \rrbracket \setminus \{j\}, \mathbf{k}}(\epsilon) + \sum_{\substack{f=1 \\ f \neq j}}^{Q} I_{\mu, \mu'}^{\mathcal{N}, \mathcal{N}', \mathcal{P}, j, \llbracket 1, Q \rrbracket \setminus \{j, f\}, \mathbf{k}}(\epsilon). \end{aligned} $$

We now conclude by using Lemma 3.1.3 by noticing the fact that $Q_{\mu,\mu'}^{1,N,N'}(\mathcal{P},j,\mathbf{k})$ is independent of $\epsilon$. We then find that the constant term of the right hand term is equal to the coefficient $Q_{\mu,\mu'}^{1,N,N'}(\mathcal{P},j,\mathbf{k})$.
# Chapter 4

## Proof of Theorem A, and of his corollaries

The aim of this chapter is to study the directional values of $Z(s, s')$ at nonpositive integers. By showing that the function $Z_{\mathbf{N},\mathbf{N}'}(s)$ is regular at $s = 0$, we have established that the quantities $Z(-(\mathbf{N}, \mathbf{N}'))$ and $Z'(-(\mathbf{N}, \mathbf{N}'))$ are well-defined. To compute the first quantity, we will use Proposition 2.2.4 to evaluate $Z_{\mathbf{N},\mathbf{N}'}(s)$ at $s = 0$.

### 4.1 Statement of Theorem A

Let's recall the statement of the theorem:

**Theorem (A).** Let $(\mu, \mu') \in \mathbb{R}_+^P \times \mathbb{R}_*^{+Q}$. Let $(\mathbf{N}, \mathbf{N}') \in \mathbb{N}_0^P \times \mathbb{N}_0^Q$. We have

$$Z(-(\mathbf{N}, \mathbf{N}')) = (-1)^{|\mathbf{N}'|} \sum_{\mathcal{P} \subseteq [1, P]} (-1)^{|\mathbf{N}|_{||\mathcal{P}||} + |\mathcal{P}|} \left( \prod_{p \in \mathcal{P}} N_p! \right) \sum_{\substack{\mathbf{k}=(k_p)_{p \in \mathcal{P}^c} \in \mathbb{N}_0^{\mathcal{P}^c} \\ |\mathbf{k}|=|\mathbf{N}'| + |\mathbf{N}|_{||\mathcal{P}||} + |\mathcal{P}|}} \sum_{j=1}^{Q} \frac{(-1)^{N'_j} \mu'_j N'_j!}{|\mu'| + |\mu|_{||\mathcal{P}||}} Q_{\mathbf{N}, \mathbf{N}'}^0(j, \mathcal{P}, \mathbf{k}) \prod_{p \in \mathcal{P}^c} \frac{\zeta(-N_p - k_p, d_p)}{k_p!},$$

with

$$Q_{\mathbf{N},\mathbf{N}'}^0(j, \mathcal{P}, \mathbf{k}) = (-1)^{|\mathbf{N}'|_{||[1,Q]\setminus\{j\}|}} \left( \prod_{\substack{q=1 \\ q \neq j}}^Q N'_q! \right) \sum_{\substack{\mathbf{w}=(\mathbf{w}_p)_{p \in \mathcal{P}^c} \in (\mathbb{N}_0^Q)^{\mathcal{P}^c} \\ \forall p \in \mathcal{P}^c |\mathbf{w}_p|=k_p \\ \forall q \neq j, \mathbf{v}(q)+\mathbf{w}(q)=N'_q}} \sum_{\substack{\mathbf{v}=(\mathbf{v}_p)_{p \in \mathcal{P}} \in (\mathbb{N}_0^{[1,Q]\setminus\{j\}})^{\mathcal{P}} \\ \forall q \neq j, \mathbf{v}(q)+\mathbf{w}(q)=N'_q}} \\ \left( \prod_{p \in \mathcal{P}^c} \binom{k_p}{\mathbf{w}_p} \prod_{q=1}^Q c_{q,p}^{w_{p,q}} \right) \left( \prod_{p \in \mathcal{P}} \binom{-N_p-1}{|\mathbf{v}_p|} \binom{|\mathbf{v}_p|}{\mathbf{v}_p} \prod_{\substack{q=1 \\ q \neq j}}^Q c_{q,p}^{v_{p,q}} \right),$$

where we noted

$$\forall q \in [1, Q], \quad \mathbf{w}(q) = \sum_{p \in \mathcal{P}^c} w_{p,q},$$

and

$$\forall q \in [1, Q] \setminus \{j\}, \quad \mathbf{v}(q) = \sum_{p \in \mathcal{P}} v_{p,q}.$$
## 4.2 Proof of Theorem A

*Proof.* By Proposition 2.2.4, we get for all $0 < \theta \ll 1$ that

$$Z_{\mathbf{N},\mathbf{N}'}(0) = K_{\mathbf{N},\mathbf{N}'}(\theta, 0) + J_{\mathbf{N},\mathbf{N}'}(\theta, 0).$$

Since $K_{\mathbf{N},\mathbf{N}'}(\theta, 0) = 0$ by Proposition 2.3.11, we only have to study the value of $J_{\mathbf{N},\mathbf{N}'}(\theta, s)$ at $s = 0$. By formula 2.5, we have

$$Z_{\mathbf{N},\mathbf{N}'}(0) = \sum_{\mathcal{P} \subseteq [1,P]} \prod_{p \in \mathcal{P}} N_p! \sum_{\mathbf{k}=(k_p)_{p \in \mathcal{P}^c} \in \mathbb{N}_0^{\mathcal{P}^c}} (-1)^{|\mathbf{k}|} \sum_{j=1}^{P} h_{\mathbf{N},\mathbf{N}',\mathcal{P},j,\mathbf{k}}(0) \left( \frac{1}{\Gamma(-N'_j + \mu'_j s)} \frac{\theta^{-|\mathbf{N}'|-|\mathbf{N}|_\mathcal{P}+(|\mu'|+|\mu|_\mathcal{P})s-|\mathcal{P}|+|\mathbf{k}|_\mathcal{P}^c}}{-|\mathbf{N}'|-|\mathbf{N}|_\mathcal{P} + (|\mu'| + |\mu|_\mathcal{P})s - |\mathcal{P}| + |\mathbf{k}|_\mathcal{P}^c} \right)_{s=0} \prod_{p \in \mathcal{P}^c} \frac{\zeta(-N_p - k_p, d_p)}{k_p!} + \left( \prod_{p=1}^P \Gamma(1+N_p) \right) \left( \sum_{j=1}^Q h_{\mathbf{N},\mathbf{N}',[1,P],j}(0) \left( \frac{\theta^{(|\mu'|+|\mu|)s-|\mathbf{N}'|-|\mathbf{N}|-P}}{\Gamma(-N'_j + \mu'_j s)((|\mu'| + |\mu|)s - |\mathbf{N}'|-|\mathbf{N}|-P)} \right)_{s=0} \right).$$

Let's first observe that the expression

$$\left( \prod_{p=1}^{P} \Gamma(1 + N_p) \right) \left( \sum_{j=1}^{Q} h_{\mathbf{N},\mathbf{N}',[1,P],j}(0) \left( \frac{\theta^{(|\mu'|+|\mu|)s-|\mathbf{N}'|-|\mathbf{N}|-P}}{\Gamma(-N'_j + \mu'_j s)((|\mu'| + |\mu|)s - |\mathbf{N}'|-|\mathbf{N}|-P)} \right)_{s=0} \right),$$

vanish because $-|\mathbf{N}'| - |\mathbf{N}| - P \neq 0$. Therefore, the zero of $\frac{1}{\Gamma(-N'_j + \mu'_j s)}$ at $s = 0$ is not compensated by any poles coming from the fraction $\frac{1}{(|\mu'| + |\mu|)s - |\mathbf{N}'| - |\mathbf{N}| - P}$ at $s = 0$.

By studying the first term of the formula of $Z_{\mathbf{N},\mathbf{N}'}(0)$, we see that $h_{\mathbf{N},\mathbf{N}',\mathcal{P},j,\mathbf{k}}(0) = Q_{\mathbf{N},\mathbf{N}'}^0(j, \mathcal{P}, \mathbf{k})$. Let $0 < \theta \ll 1$ and $\mathcal{P} \subseteq [1, P]$.

* If $|\mathbf{N}'| + |\mathbf{N}|_\mathcal{P} \neq -|\mathcal{P}| + |\mathbf{k}|_\mathcal{P}^c$, then by Lemma 3.1.1 we find that

$$\frac{1}{\Gamma(-N'_j + \mu'_j s)} \frac{\theta^{-|\mathbf{N}'|-|\mathbf{N}|_\mathcal{P}+(|\mu'|+|\mu|_\mathcal{P})s-|\mathcal{P}|+|\mathbf{k}|_\mathcal{P}^c}}{-|\mathbf{N}'|-|\mathbf{N}|_\mathcal{P} + (|\mu'| + |\mu|_\mathcal{P})s - |\mathcal{P}| + |\mathbf{k}|_\mathcal{P}^c}$$

vanish at $s = 0$.

* If $|\mathbf{N}'| + |\mathbf{N}|_\mathcal{P} = -|\mathcal{P}| + |\mathbf{k}|_\mathcal{P}^c$, then by Lemma 3.1.1 we find that:

$$\left( \frac{1}{\Gamma(-N'_j + \mu'_j s)} \frac{\theta^{-|\mathbf{N}'|-|\mathbf{N}|_\mathcal{P}+(|\mu'|+|\mu|_\mathcal{P})s-|\mathcal{P}|+|\mathbf{k}|_\mathcal{P}^c}}{-|\mathbf{N}'|-|\mathbf{N}|_\mathcal{P} + (|\mu'| + |\mu|_\mathcal{P})s - |\mathcal{P}| + |\mathbf{k}|_\mathcal{P}^c} \right)_{s=0} = \frac{\mu'_j}{|\mu'| + |\mu|_\mathcal{P}} (-1)^{N'_j} N_j^{N'_j},$$

therefore we prove the Theorem A. The expression of $Q_{\mathbf{N},\mathbf{N}'}^0(j, \mathcal{P}, \mathbf{k})$ in Theorem A is easily deduced by the formula (3.6). □

## 4.3 On corollaries of Theorem A

We easily derive from Theorem A his corollaries.
### 4.3.1 Proof of Corollary A1

To prove Corollary A1, we apply Theorem A by setting $(\mathbf{N}, \mathbf{N}') = (\mathbf{0}, \mathbf{0})$:

$$Z(-(\mathbf{0}, \mathbf{0})) = \sum_{\mathcal{P} \subseteq [1, P]} (-1)^{|\mathcal{P}|} \sum_{\mathbf{k}=(k_p)_{p \in \mathcal{P}} \in \mathbb{N}_0^{\mathcal{P}^c} \atop |\mathbf{k}| = |\mathcal{P}|} \sum_{j=1}^{Q} \frac{\mu'_j}{|\mathbf{\mu}'| + |\mathbf{\mu}|_{\mathcal{P}}} Q_{\mathbf{0},0}^{0}(j, \mathcal{P}, \mathbf{k}) \prod_{p \in \mathcal{P}^c} \frac{\zeta(-k_p, d_p)}{k_p!},$$

with

$$Q_{\mathbf{0},0}^{0}(j, \mathcal{P}, \mathbf{k}) = \sum_{\substack{\mathbf{w}=(\mathbf{w}_p)_{p \in \mathcal{P}} \in (\mathbb{N}_0^Q)^{\mathcal{P}^c} \\ \forall p \in \mathcal{P}^c | \mathbf{w}_p| = k_p \\ \sum_{q \neq j, \mathbf{v}(q) + \mathbf{w}(q) = 0}} \sum_{\substack{\mathbf{v}=(\mathbf{v}_p)_{p \in \mathcal{P}} \in (\mathbb{N}_0^{[1,Q]} \setminus \{j\})^\mathcal{P} \\ \forall q \neq j, \mathbf{v}(q) + \mathbf{w}(q) = 0}} \left( \prod_{p \in \mathcal{P}^c} \binom{k_p}{\mathbf{w}_p} \prod_{q=1}^{Q} c_{q,p}^{w_p,q} \right) \left( \prod_{p \in \mathcal{P}} \binom{-1}{|\mathbf{v}_p|} \binom{|\mathbf{v}_p|}{\mathbf{v}_p} \prod_{\substack{q=1 \\ q \neq j}}^{Q} c_{q,p}^{v_p,q} \right).$$

We can simplify the coefficient $Q_{\mathbf{0},0}^{0}(j, \mathcal{P}, \mathbf{k})$ by noticing that the condition $\mathbf{v}(q) + \mathbf{w}(q) = 0$ for all $q \neq j$ implies that

$$\forall p \in \mathcal{P}, \forall q \in [1, Q] \setminus \{j\}, \quad v_{p,q} = 0$$

$$\forall p \in \mathcal{P}^c, \forall q \in [1, Q] \setminus \{j\}, \quad w_{p,q} = 0.$$

We then obtain that $k_p = |\mathbf{w}_p| = w_{p,j}$ for all $p \in \mathcal{P}^c$. Therefore we have the following expression for $Q_{\mathbf{0},0}^{0}(j, \mathcal{P}, \mathbf{k})$:

$$Q_{\mathbf{0},0}^{0}(j, \mathcal{P}, \mathbf{k}) = \prod_{p \in \mathcal{P}^c} c_{j,p}^{k_p}.$$

### 4.3.2 Proof of Corollary A2

On Corollary A2, we set $(\mu, \mu') = (\mathbf{1}, \mathbf{1})$ and $(\mathbf{N}, \mathbf{N}') = (N, \dots, N)$, and we find by Theorem A that

$$Z_{\Delta}(-N) = \sum_{\mathcal{P} \subseteq [1, P]} (-1)^{N(|\mathcal{P}|+Q+1)+|\mathcal{P}|N!|\mathcal{P}|+1} \sum_{\mathbf{k}=(k_p)_{p \in \mathcal{P}} \in \mathbb{N}_0^{\mathcal{P}^c} \atop |\mathbf{k}|=N(Q+|\mathcal{P}|)+|\mathcal{P}|} \sum_{j=1}^{Q} \frac{1}{Q + |\mathcal{P}|} Q_N^0(j, \mathcal{P}, \mathbf{k}) \prod_{p \in \mathcal{P}^c} \frac{\zeta(-N - k_p, d_p)}{k_p!},$$

with

$$Q_N^0(j, \mathcal{P}, \mathbf{k}) = (-1)^{N(Q-1)} N!^{Q-1} \sum_{\substack{\mathbf{w}=(\mathbf{w}_p)_{p \in \mathcal{P}} \in (\mathbb{N}_0^Q)^{\mathcal{P}^c} \\ \forall p \in \mathcal{P}^c | \mathbf{w}_p| = k_p \\ \sum_{\substack{\mathbf{v}=(\mathbf{v}_p)_{p \in \mathcal{P}} \in (\mathbb{N}_0^{[1,Q]} \setminus \{j\})^\mathcal{P} \\ \forall q \neq j, \mathbf{v}(q) + \mathbf{w}(q) = N}} \left( \prod_{p \in \mathcal{P}^c} \binom{k_p}{\mathbf{w}_p} \prod_{q=1}^{Q} c_{q,p}^{w_p,q} \right) \left( \prod_{p \in \mathcal{P}} \binom{-N-1}{|\mathbf{v}_p|} \binom{|\mathbf{v}_p|}{\mathbf{v}_p} \prod_{\substack{q=1 \\ q \neq j}}^{Q} c_{q,p}^{v_p,q} \right),$$

where we noted

$$Q_N^0(\mathcal{P}, \mathbf{k}) = \sum_{j=1}^{Q} Q_N^0(j, \mathcal{P}, \mathbf{k}),$$

therefore proving the required result.
### 4.3.3 Proof of Corollary A3

The Corollary A3 corresponds to a qualitative and non explicit version of Theorem A. Let $(\boldsymbol{\mu}, \boldsymbol{\mu}') \in \mathbb{R}_+^P \times \mathbb{R}_*^{Q^+}$. Let $(\mathbf{N}, \mathbf{N}') \in \mathbb{N}_0^P \times \mathbb{N}_0^Q$. We define $\mathbb{K}$ the field above $\mathbb{Q}$ generated by the complex numbers $(d_p)_{p \in [1, P]}$ and $(c_{q,p})_{p \in [1, P], q \in [1, Q]}$, and by the directions $\mu_p$ for all $p \in [1, P]$, and $\mu'_q$ for all $q \in [1, Q]$.
By Theorem A, we find that

$$ Z(-(\mathbf{N}, \mathbf{N}')) = (-1)^{|\mathbf{N}'|} \sum_{\mathcal{P} \subseteq [1, P]} (-1)^{|\mathbf{N}_{||\mathcal{P}}| + |\mathcal{P}|} \left( \prod_{p \in \mathcal{P}} N_p! \right) \sum_{\substack{\mathbf{k}=(k_p)_{p \in \mathcal{P}} \in \mathbb{N}_0^{\mathcal{P}^c} \\ |\mathbf{k}|=|\mathbf{N}'|+|\mathbf{N}_{||\mathcal{P}}|+|\mathcal{P}|}} \sum_{j=1}^{Q} \frac{(-1)^{N'_j} \mu'_j N'_j!}{|\boldsymbol{\mu}'| + |\boldsymbol{\mu}_{||\mathcal{P}}|} Q_{\mathbf{N}, \mathbf{N}'}^0(j, \mathcal{P}, \mathbf{k}) \\ \prod_{p \in \mathcal{P}^c} \frac{\zeta(-N_p - k_p, d_p)}{k_p!}, $$

with

$$ Q_{\mathbf{N}, \mathbf{N}'}^0(j, \mathcal{P}, \mathbf{k}) = (-1)^{|\mathbf{N}'|_{[1,Q]\setminus\{j\}}} \left( \prod_{\substack{q=1 \\ q \neq j}}^Q N'_q! \right) \sum_{\substack{\mathbf{w}=(\mathbf{w}_p)_{p \in \mathcal{P}} \in (\mathbb{N}_0^Q)^{\mathcal{P}^c} \\ \forall p \in \mathcal{P}^c |\mathbf{w}_p|=k_p \\ \mathbf{v}=(\mathbf{v}_p)_{p \in \mathcal{P}} \in (\mathbb{N}_0^{[1,Q]\setminus\{j\}})^{\mathcal{P}} \\ \forall q \neq j, \mathbf{v}(q)+\mathbf{w}(q)=N'_q}} \sum_{\substack{\mathbf{v}=(\mathbf{v}_p)_{p \in \mathcal{P}} \in (\mathbb{N}_0^{[1,Q]\setminus\{j\}})^{\mathcal{P}} \\ \forall q \neq j, \mathbf{v}(q)+\mathbf{w}(q)=N'_q}} \\ \left( \prod_{p \in \mathcal{P}^c} \binom{k_p}{\mathbf{w}_p} \prod_{q=1}^Q c_{q,p}^{w_{p,q}} \right) \left( \prod_{p \in \mathcal{P}} \binom{-N_p-1}{|\mathbf{v}_p|} \binom{|\mathbf{v}_p|}{|\mathbf{w}_p|} \prod_{\substack{q=1 \\ q \neq j}}^Q c_{q,p}^{v_{p,q}} \right). $$

Let's now observe that the coefficients $Q_{\mathbf{N}, \mathbf{N}'}^0(j, \mathcal{P}, \mathbf{k})$ belong to the field $\mathbb{K}$. Moreover, by property of the special values of the Hurwitz zeta function, we know that the nonpositive values of this zeta function have an expression using only Bernoulli polynomials, which are all rational polynomials. Therefore we get that $\frac{\zeta(-N_p - k_p, d_p)}{k_p!} \in \mathbb{Q}[d_1, \dots, d_P] \subset \mathbb{K}$, which proves the corollary.
# Chapter 5

## Proof of Proposition B, Theorem D, and some corollaries

### 5.1 Statement of Proposition B and of Theorem D

Firstly we will prove in this chapter Proposition B, and secondly we shall prove Theorem D, which we recall the statements below:

**Proposition (B).** Let $j \in [1, Q]$ an positive integer. We assume that $l_j$ is a linear form with rational coefficients for all $1 \le j \le Q$. For all $1 \le p \le P$, we then write $c_{j,p} = \frac{a_{j,p}}{b_{j,p}}$ the irreducible decomposition of $c_{j,p}$ with $a_{j,p} > 0$ and $b_{j,p} > 0$. Let $x_j(\mathbf{c}) = \frac{\text{lcm}(a_{j,p})}{\text{gcd}(b_{j,p})} \in \mathbb{Q}_+^*$, and $\beta_{j,p}(\mathbf{c}) := \frac{x_j(\mathbf{c})}{c_{j,p}} \in \mathbb{N}$ for all $1 \le p \le P$. Then, for all $\mathbf{R} \in \mathbb{N}_0^P$, we have for all $s \in \mathbb{C} \setminus \mathbb{N}$,

$$ \varphi_{\mathbf{R}}^j(s) = x_j(\mathbf{c})^{-s} \prod_{p=1}^{P} \beta_{j,p}^{R_p} \sum_{\substack{0 \le v_1 \le \beta_{j,1}(\mathbf{c})-1 \\ 0 \le v_P \le \beta_{j,P}(\mathbf{c})-1}} \sum_{\mathcal{P} \subseteq [1,P]} \prod_{p \in \mathcal{P}^c} R_p! \sum_{\substack{\mathbf{k} \in \mathbb{N}_0^P, k' \in \mathbb{N}_0 \\ |\mathbf{k}|+k'=|\mathbf{R}|+|\mathcal{P}^c|+1}} (-1)^{|\mathbf{k}|} \prod_{p \in \mathcal{P}} \frac{\zeta\left(-R_p - k_p, \frac{d_p+v_p}{\beta_{j,p}(\mathbf{c})}\right)}{k_p!} \cdot \frac{\zeta(s-k', l_j(x_j(\mathbf{c}))^{-1}(\mathbf{d}+\mathbf{v})))}{k'!} $$
**Theorem (D).** Let $(\mathbf{N}, \mathbf{N}') \in \mathbb{N}_0^P \times \mathbb{N}_0^Q$, and $(\boldsymbol{\mu}, \boldsymbol{\mu}') \in \mathbb{R}^{+P} \times \mathbb{R}_*^{+Q}$, then

$$Z'(-(\mathbf{N}, \mathbf{N}')) = \tag{5.1}$$

$$ \begin{aligned} & \sum_{\mathcal{P} \subseteq [1,P]} \sum_{j=1}^{Q} (-1)^{|\mathbf{N}|+|\mathbf{N}'|+|\mathcal{P}|+N'_j} \left( \prod_{p \in \mathcal{P}} N_p! \right) \frac{\mu'_j N'_j!}{|\boldsymbol{\mu}'| + |\boldsymbol{\mu}|_{\mathcal{P}}} \sum_{\substack{\mathbf{k}=(k_p)_{p \in \mathcal{P}} \in \mathbb{N}_0^{\mathcal{P}^c} \\ |\mathbf{k}|=|\mathbf{N}'|+|\mathbf{N}|_{\mathcal{P}}+|\mathcal{P}|}} \prod_{p \in \mathcal{P}} \frac{\zeta(-N_p - k_p, d_p)}{k_p!} \\ & \qquad \cdot \left( Q_{\mathbf{N},\mathbf{N}'}^0(j, \mathcal{P}, \mathbf{k}) \left( \sum_{p \in \mathcal{P}} \mu_p(\gamma - h_{N_p}) + \mu'_j(\gamma - h_{N'_j}) \right) + Q_{\mathbf{N},\mathbf{N}'}^1(\boldsymbol{\mu}, \boldsymbol{\mu}') \right) \\ & \qquad + \sum_{\mathcal{P} \subseteq [1,P]} \sum_{j=1}^{Q} \left( \prod_{p \in \mathcal{P}} N_p! \right) (-1)^{|\mathbf{N}'|+|\mathbf{N}|_{\mathcal{P}}+|\mathcal{P}|+N'_j} \frac{\mu'_j N'_j!}{|\boldsymbol{\mu}'| + |\boldsymbol{\mu}|_{\mathcal{P}}} \sum_{\substack{\mathbf{k}=(k_p)_{p \in \mathcal{P}} \in \mathbb{N}_0^{\mathcal{P}^c} \\ |\mathbf{k}|=|\mathbf{N}'|+|\mathbf{N}|_{\mathcal{P}}+|\mathcal{P}|}} Q_{\mathbf{N},\mathbf{N}'}^0(j, \mathcal{P}, \mathbf{k}) \\ & \qquad \cdot \left( \sum_{i \in \mathcal{P}^c} \mu_i \frac{\zeta'(-N_i - k_i, d_i)}{k_i!} \prod_{p \in \mathcal{P}^c \setminus \{i\}} \frac{\zeta(-N_p - k_p, d_p)}{k_p!} \right) \\ & \qquad + \sum_{j=1}^{Q} \mu'_j \sum_{\substack{\mathbf{u}=(\mathbf{u}_q)_{q \in [1,Q]\setminus\{j\}} \in (\mathbb{N}_0^P)^{[1,Q]\setminus\{j\}} \\ \forall q \in [1,Q]\setminus\{j\}, |\mathbf{u}_q|=N'_q}} \left( \prod_{q=1}^{Q} \binom{N'_q}{q \neq j} \prod_{p=1}^{P} c_{q,p}^{u_{q,p}} \right) \partial_s \varphi_{\mathbf{R}(\mathbf{N},j,\mathbf{u})}^j(s)|_{s=-N'_j} \\ & \qquad - \sum_{j=1}^{Q} (\gamma - h_{N'_j}) \mu'_j N'_j! \sum_{\substack{\mathbf{u}=(\mathbf{u}_q)_{q \in [1,Q]\setminus\{j\}} \in (\mathbb{N}_0^P)^{[1,Q]\setminus\{j\}} \\ \forall q \in [1,Q]\setminus\{j\}, |\mathbf{u}_q|=N'_q}} \left( \left( \prod_{q=1}^{Q} \binom{N'_q}{q \neq j} \prod_{p=1}^{P} c_{q,p}^{u_{q,p}} \right) \partial_s \varphi_{\mathbf{R}(\mathbf{N},j,\mathbf{u})}^j(s) \right) \end{aligned} $$

with

$$\mathbf{R}(\mathbf{N}, j, \mathbf{u}) := (N_p + \mathbf{u}(p))_{p \in [1,P]},$$

for all $\mathbf{u} = (\mathbf{u}_q)_{q \in [1,Q]\setminus\{j\}}$, and for all $j \in [1, Q]$,

$$ Q_{\mathbf{N},\mathbf{N}'}^0(j, \mathcal{P}, \mathbf{k}) = \sum_{\substack{\mathbf{w}=(\mathbf{w}_p)_{p \in \mathcal{P}} \in (\mathbb{N}_0^Q)^{\mathcal{P}^c} \\ \forall p \in \mathcal{P}^c |\mathbf{w}_p|=k_p}} \sum_{\substack{\mathbf{v}=(\mathbf{v}_p)_{p \in \mathcal{P}} \in (\mathbb{N}_0^{[1,Q]} \setminus \{j\})^P \\ \forall q \in [1,Q]\setminus\{j\}, N'_q=\mathbf{v}(q)+\mathbf{w}(q)}} \left( \prod_{p \in \mathcal{P}} \binom{k_p}{\mathbf{w}_p} \prod_{q=1}^{Q} c_{q,p}^{w_{p,q}} \right) \\ \cdot \left( \prod_{p \in \mathcal{P}} \binom{-N_p-1}{|\mathbf{v}_p|} \binom{|\mathbf{v}_p|}{\mathbf{v}_p} \prod_{\substack{q=1 \\ q \neq j}}^{Q} c_{q,p}^{v_{p,q}} \right) \prod_{\substack{q=1 \\ q \neq j}}^{Q} (-1)^{N'_q} N'_q!, $$
and

$$ Q_{\mu, \mu'}^{1, \mathbf{N}, \mathbf{N}'}(j, \mathcal{P}, \mathbf{k}) = (-1)^{|\mathbf{N}'| \cap \{1, \dots, j\}} \prod_{\substack{q=1 \\ q \neq j}}^{Q} N_q'! \sum_{\substack{\mathbf{w}=(\mathbf{w}_p)_{p \in \mathcal{P}} \in (\mathbb{N}_0^Q)^{\mathcal{P}} \\ \forall p \in \mathcal{P}^c | \mathbf{w}_p| = k_p}} \sum_{\substack{\mathbf{v}=(\mathbf{v}_p)_{p \in \mathcal{P}} \in (\mathbb{N}_0^{[1,Q]} \setminus \{j\})^{\mathcal{P}} \\ \forall q \neq j, \mathbf{v}(q) + \mathbf{w}(q) = N_q' \\ \forall p \in \mathcal{P}^c | \mathbf{w}_p| = k_p}} \left( \prod_{p \in \mathcal{P}^c} \binom{k_p}{\mathbf{w}_p} \prod_{q=1}^{Q} c_{q,p}^{w_{p,q}} \right) \left( \prod_{p \in \mathcal{P}} \binom{-N_p - 1}{|\mathbf{v}_p|} \binom{|\mathbf{v}_p|}{|\mathbf{v}_p|} \prod_{\substack{q=1 \\ q \neq j}}^{Q} c_{q,p}^{w_{p,q}} \right) \left( \sum_{p \in \mathcal{P}} \mu_p (h_{N_p} - h_{N_p+|\mathbf{v}_p|}) + \sum_{\substack{q=1 \\ q \neq j}}^{Q} \mu'_q (\gamma - h_{N'_q}) \right) \\ + (-1)^{|\mathbf{N}'| \cap \{1, \dots, j\}} \left( \prod_{\substack{q=1 \\ q \neq j}}^{Q} N_q'! \right) \sum_{\substack{f=1 \\ f \neq j}}^{Q} \mu'_f \sum_{\substack{\mathbf{w}=(\mathbf{w}_p)_{p \in \mathcal{P}} \in (\mathbb{N}_0^Q)^{\mathcal{P}} \\ \forall p \in \mathcal{P}^c | \mathbf{w}_p| = k_p}} \sum_{\substack{\mathbf{v}=(\mathbf{v}_p)_{p \in \mathcal{P}} \in (\mathbb{N}_0^{[1,Q]} \setminus \{j,f\})^{\mathcal{P}} \\ \forall q \neq j, f, \mathbf{v}(q) + \mathbf{w}(q) = N'_q \\ \forall p \in \mathcal{P}^c | \mathbf{w}_p| = k_p}} \left( \prod_{p \in \mathcal{P}} \binom{k_p}{\mathbf{w}_p} \prod_{q=1}^{Q} c_{q,p}^{w_{p,q}} \right) \left( \prod_{p \in \mathcal{P}} \binom{-N_p - 1}{|\mathbf{v}_p|} \binom{|\mathbf{v}_p|}{|\mathbf{v}_p|} \prod_{\substack{q=1 \\ q \neq j, f}}^{Q} c_{q,p}^{w_{p,q}} \right) F_{\mathcal{P}, j, f, \mathbf{v}, \mathbf{w}}(\mathbf{N}, \mathbf{N}'), $$

with $F_{\mathcal{P},j,f,\mathbf{v},\mathbf{w}}(\mathbf{N}, \mathbf{N}')$ being the constant described in Proposition 1.3.2.

## 5.2 Auxiliary functions

In this section, we shall apply to the auxiliary functions $\varphi_{\mathbf{R}}^j(s)$ the Proposition 2.2.4 in order to use these formulas when we will compute the special values of the derivative $Z'(-(\mathbf{N}, \mathbf{N}'))$.

Let's recall that, for $\mathbf{R} = (R_1, \dots, R_P) \in \mathbb{N}_0^P$ and $j \in [1, Q]$, we have set

$$ \varphi_{\mathbf{R}}^{j}(s) = \sum_{\mathbf{n} \in \mathbb{N}_{0}^{P}} (n_{1} + d_{1})^{R_{1}} \dots (n_{P} + d_{P})^{R_{P}} (l_{j}(\mathbf{n}) + d'_{j})^{-s}. $$

This Dirichlet series is meromorphic on $\mathbb{C}$, and we know that the auxiliary function is regular at nonpositive integers by Corollary 2.2.6.

**Remark 5.2.1.** For $\mathbf{R} = \mathbf{0}$, we retrieve the Barnes zeta function defined as

$$ \zeta^B(s, d'_j | c_{j,1}, \dots, c_{j,P}) = \sum_{\mathbf{n} \in \mathbb{N}_0^P} \frac{1}{(l_j(\mathbf{n}) + d'_j)^s}. $$

We apply Proposition 2.2.4 to the auxiliary function $\varphi_{\mathbf{R}}^j(s)$, setting $\boldsymbol{\mu} = \mathbf{0}$, $\mathbf{N} = \mathbf{R}$, $\mathbf{N}' = \mathbf{0}$, and $\boldsymbol{\mu}' = \mathbf{1}$. We know that this auxiliary function is regular at nonpositive integers, and by this same proposition, we obtain that the set of poles is contained in the set

$$ S(\mathbf{R}) := \mathbb{Z}_{\ge 1}. $$

We find that the auxiliary function takes the following form, for $0 < \theta \ll 1$, and for all $s \in \mathbb{C} \setminus S(\mathbf{R})$,

$$ \varphi_{\mathbf{R}}^{j}(s) = J_{\mathbf{R}}^{j}(\theta, s) + K_{\mathbf{R}}^{j}(\theta, s), \qquad (5.2) $$

such that for all $s \in \mathbb{C}$,

$$ K_{\mathbf{R}}^{j}(\theta, s) = \frac{1}{\Gamma(s)} \sum_{\mathbf{n} \in \mathbb{N}_{0}^{P}} \Gamma(s, \theta, l_{j}(\mathbf{n}) + d'_{j}) \prod_{p=1}^{P} (n_{p} + d_{p})^{R_{p}}, $$
and for all $s \in \mathbb{C} \setminus S(\mathbf{R})$, we have

$$
J_{\mathbf{R}}^{j}(\theta, s) = \sum_{\mathcal{P} \subseteq [1, P]} \prod_{p \in \mathcal{P}} R_p! \sum_{\mathbf{k}=(k_p)_{p \in \mathcal{P}^c} \in \mathbb{N}_0^{\mathcal{P}^c}} (-1)^{|\mathbf{k}|} h_{\mathbf{R}, \mathcal{P}, \mathbf{k}}(s) \frac{\theta^{s-|\mathbf{R}|_{\mathcal{P}}-|\mathcal{P}|+|\mathbf{k}|}}{\Gamma(s)(s-|\mathbf{R}|_{\mathcal{P}}-|\mathcal{P}|+|\mathbf{k}|)} \prod_{p \in \mathcal{P}^c} \frac{\zeta(-R_p - k_p, d_p)}{k_p!} \\
+ \left(\prod_{p=1}^{P} R_p!\right) h_{\mathbf{R}, [1, P], j}(s) \frac{\theta^{|\mu'|+|\mu|} s - |N'|-|N|-P}{\Gamma(s)(s-|\mathbf{R}|-P)},
$$

with

$$
h_{\mathbf{R}, \mathcal{P}, \mathbf{k}}(s) = \prod_{p \in \mathcal{P}} c_{j,p}^{-N_p-1} \prod_{p \in \mathcal{P}^c} c_{j,p}^{k_p},
$$

and

$$
h_{\mathbf{R}, [1, P], j}(s) = \prod_{p=1}^{P} c_{j,p}^{-N_p-1}.
$$

We then get

$$
J_{\mathbf{R}}^{j}(\theta, s) = \tag{5.3}
\sum_{\mathcal{P} \subseteq [1, P]} \prod_{p \in \mathcal{P}} R_p! \sum_{\mathbf{k}=(k_p)_{p \in \mathcal{P}^c} \in \mathbb{N}_0^{\mathcal{P}^c}} (-1)^{|\mathbf{k}|} \left( \prod_{p \in \mathcal{P}} c_{j,p}^{-R_p-1} \prod_{p \in \mathcal{P}^c} c_{j,p}^{k_p} \right) \frac{\theta^{s-|\mathbf{R}|_{\mathcal{P}}-|\mathcal{P}|+|\mathbf{k}|}}{\Gamma(s)(s-|\mathbf{R}|_{\mathcal{P}}-|\mathcal{P}|+|\mathbf{k}|)} \prod_{p \in \mathcal{P}^c} \frac{\zeta(-R_p - k_p, d_p)}{k_p!} \\
+ \left( \prod_{p=1}^{P} R_p! \right) \left( \prod_{p=1}^{P} c_{j,p}^{-R_p-1} \right) \frac{\theta^{s-|\mathbf{R}|-P}}{\Gamma(s)(s-|\mathbf{R}|-P)}.
$$

**Proposition 5.2.2.** Let $\mathbf{R} \in \mathbb{N}_0^P$, and $N \in \mathbb{N}_0$. Let $0 < \theta \ll 1$ and $j \in [1, Q]$, then $s \mapsto \varphi_{\mathbf{R}}^j(s)$ is meromorphic on $\mathbb{C}$, and is regular at nonpositive integers. Moreover, we have

$$
\partial_s K_{\mathbf{R}}^j(\theta, -N) = (-1)^N N! \sum_{\mathbf{n} \in \mathbb{N}_0^P} \Gamma(-N, \theta, l_j(\mathbf{n}) + d_j') \prod_{p=1}^P (n_p + d_p)^{R_p} \quad (5.4)
$$

$$
\partial_s J_{\mathbf{R}}^j(\theta, -N) = \tag{5.5}
\sum_{\mathcal{P} \subseteq [1, P]} (-1)^{|\mathbf{R}|_{\mathcal{P}} + |\mathcal{P}|} \prod_{p \in \mathcal{P}} c_{j,p}^{-R_p-1} R_p! \sum_{\substack{\mathbf{k}=(k_p)_{p \in \mathcal{P}^c} \in \mathbb{N}_0^{\mathcal{P}^c} \\ |\mathbf{k}|=N+|\mathbf{R}|_{\mathcal{P}}+|\mathcal{P}|}} \prod_{p \in \mathcal{P}^c} c_{j,p}^{k_p} \frac{\zeta(-R_p - k_p, d_p)}{k_p!},
$$

with $J_{\mathbf{R}}^{*j}(\theta, -N)$ being a function of the form $J_{\mathbf{R}}^{*j}(\theta, -N) = \sum_{n \ge -N - |\mathbf{R}| - P} \alpha_n \theta^n + \beta \ln(\theta)$ for all $0 < \theta \ll 1$, and without any constant term (i.e. $\alpha_0 = 0$).

**Proof.** By Theorem 2.2.6, we know that $\varphi_{\mathbf{R}}^j$ is meromorphic at nonpositive integers. By Proposition 2.3.11, we know that the function $s \in \mathbb{C} \mapsto \sum_{\mathbf{n} \in \mathbb{N}_0^P} \Gamma(s, \theta, l_j(\mathbf{n}) + d_j') \prod_{p=1}^P (n_p + d_p)^{R_p}$ is holomorphic on $\mathbb{C}$.

Deriving $K_{\mathbf{R}}^j$ with respect to $s$, we get

$$
\partial_s K_{\mathbf{R}}^j(\theta, s) = - \frac{\psi(s)}{\Gamma(s)} \sum_{\mathbf{n} \in \mathbb{N}_0^P} \Gamma(s, \theta, l_j(\mathbf{n}) + d_j') \prod_{p=1}^P (n_p + d_p)^{R_p} \\
+ \frac{1}{\Gamma(s)} \partial_s \left( \sum_{\mathbf{n} \in \mathbb{N}_0^P} \Gamma(s, \theta, l_j(\mathbf{n}) + d_j') \prod_{p=1}^P (n_p + d_p)^{R_p} \right),
$$
where $\psi$ is the digamma function.

Therefore we can evaluate the sum of these two terms at $s = -N$. Since $\frac{1}{\Gamma(s)}$ vanish at $s = -N$, and

by Lemma 3.1.1, we have $\left(\frac{\psi(s)}{\Gamma(s)}\right)_{s=-N} = (-1)^{N+1}N!$, we then obtain the required formula.

From formula 5.3, we get

$$ \partial_s J_{\mathbf{R}}^j(\theta, s) = \sum_{\mathcal{P} \subseteq [1, P]} \prod_{p \in \mathcal{P}} c_{j,p}^{-R_p-1} R_p! \sum_{\mathbf{k}=(k_p)_{p \in \mathcal{P}} \in \mathbb{N}_0^{\mathcal{P}^c}} (-1)^{|\mathbf{k}|} \prod_{p \in \mathcal{P}^c} c_{j,p}^{k_p} \prod_{p \in \mathcal{P}^c} \frac{\zeta(-R_p - k_p, d_p)}{k_p!} \\ \cdot \partial_s \left( \frac{1}{\Gamma(s)(s - |\mathbf{R}|_{\mathcal{P}} - |\mathcal{P}| + |\mathbf{k}|)} \right) \theta^{s - |\mathbf{R}|_{\mathcal{P}} - |\mathcal{P}| + |\mathbf{k}|} \\ + \sum_{\mathcal{P} \subseteq [1, P]} \prod_{p \in \mathcal{P}} c_{j,p}^{-R_p-1} R_p! \sum_{\mathbf{k}=(k_p)_{p \in \mathcal{P}} \in \mathbb{N}_0^{\mathcal{P}^c}} (-1)^{|\mathbf{k}|} \prod_{p \in \mathcal{P}^c} c_{j,p}^{k_p} \prod_{p \in \mathcal{P}^c} \frac{\zeta(-R_p - k_p, d_p)}{k_p!} \\ \cdot \frac{1}{\Gamma(s)(s - |\mathbf{R}|_{\mathcal{P}} - |\mathcal{P}| + |\mathbf{k}|)} \theta^{s - |\mathbf{R}|_{\mathcal{P}} - |\mathcal{P}| + |\mathbf{k}|} \ln(\theta) \\ - \left( \prod_{p=1}^P R_p! \right) \left( \prod_{p=1}^P c_{j,p}^{-R_p-1} \right) \theta^{s - |\mathbf{R}| - P} \left( \frac{\ln(\theta)}{\Gamma(s)(s - |\mathbf{R}| - P)} + \partial_s \left( \frac{1}{\Gamma(s)(s - |\mathbf{R}| - P)} \right) \right). $$

In order to obtain a formula for $(\partial_s J_{\mathbf{R}}^j(\theta, s))_{s=-N}$, we first need to evaluate the terms $\frac{1}{(s - |\mathbf{R}|_{\mathcal{P}} - |\mathcal{P}| + |\mathbf{k}|)\Gamma(s)}$, and the terms $\partial_s \frac{1}{(s - |\mathbf{R}|_{\mathcal{P}} - |\mathcal{P}| + |\mathbf{k}|)\Gamma(s)}$. That can easily be done using Lemma 3.1.1:

$$ \left( \frac{1}{(s - |\mathbf{R}|_{\mathcal{P}} - |\mathcal{P}| + |\mathbf{k}|)\Gamma(s)} \right)_{s=-N} = \begin{cases} (-1)^N N! & \text{if } |\mathbf{k}| = N + |\mathbf{R}|_{\mathcal{P}} + |\mathcal{P}| \\ 0 & \text{if } |\mathbf{k}| \neq N + |\mathbf{R}|_{\mathcal{P}} + |\mathcal{P}|. \end{cases} $$

and

$$ \partial_s \left( \frac{1}{\Gamma(s)(s - |\mathbf{R}|_{\mathcal{P}} - |\mathcal{P}| + |\mathbf{k}|)} \right)_{s=-N} = \begin{cases} (-1)^N (\gamma - h_N) N! & \text{if } |\mathbf{k}| = N + |\mathbf{R}|_{\mathcal{P}} + |\mathcal{P}| \\ \frac{(-1)^N N!}{-N - |\mathbf{R}|_{\mathcal{P}} - |\mathcal{P}| + |\mathbf{k}|} & \text{if } |\mathbf{k}| \neq N + |\mathbf{R}|_{\mathcal{P}} + |\mathcal{P}|. \end{cases} $$

Therefore we find that

$$ \begin{aligned} \partial_s J_{\mathbf{R}}^j(\theta, -N) = & (\gamma - h_N)N! \sum_{\mathcal{P} \subseteq [1, P]} (-1)^{|\mathbf{R}|_{\mathcal{P}} + |\mathcal{P}|} \prod_{p \in \mathcal{P}} c_{j,p}^{-R_p-1} R_p! \sum_{\substack{\mathbf{k}=(k_p)_{p \in \mathcal{P}} \in \mathbb{N}_0^{\mathcal{P}^c} \\ |\mathbf{k}|=N+|\mathbf{R}|_{\mathcal{P}}+|\mathcal{P}|}} (-1)^{|\mathbf{k}|} \prod_{p \in \mathcal{P}^c} c_{j,p}^{k_p} \frac{\zeta(-R_p - k_p, d_p)}{k_p!} \\ & + (-1)^N N! \sum_{\mathcal{P} \subseteq [1, P]} \prod_{p \in \mathcal{P}} c_{j,p}^{-R_p-1} R_p! \sum_{\substack{\mathbf{k}=(k_p)_{p \in \mathcal{P}} \in \mathbb{N}_0^{\mathcal{P}^c} \\ |\mathbf{k}| \neq N+|\mathbf{R}|_{\mathcal{P}}+|\mathcal{P}|}} (-1)^{|\mathbf{k}|} \left( \prod_{p \in \mathcal{P}^c} c_{j,p}^{k_p} \frac{\zeta(-R_p - k_p, d_p)}{k_p!} \right) \frac{\theta^{-N-|\mathbf{R}|_{\mathcal{P}}-|\mathcal{P}|+|\mathbf{k}|}}{-N-|\mathbf{R}|_{\mathcal{P}}-|\mathcal{P}|+|\mathbf{k}|} \\ & + N! \sum_{\mathcal{P} \subseteq [1, P]} (-1)^{|\mathbf{R}|_{\mathcal{P}} + |\mathcal{P}|} \prod_{p \in \mathcal{P}} c_{j,p}^{-R_p-1} R_p! \sum_{\substack{\mathbf{k}=(k_p)_{p \in \mathcal{P}} \in \mathbb{N}_0^{\mathcal{P}^c} \\ |\mathbf{k}|=N+|\mathbf{R}|_{\mathcal{P}}+|\mathcal{P}|}} \left( \prod_{p \in \mathcal{P}^c} c_{j,p}^{k_p} \frac{\zeta(-R_p - k_p, d_p)}{k_p!} \right) \ln(\theta) \\ & + (-1)^N N! \left( \prod_{p=1}^P R_p! \right) \left( \prod_{p=1}^P c_{j,p}^{-R_p-1} \right) \frac{\theta^{-N-|\mathbf{R}|-P}}{-N-|\mathbf{R}|-P}. \end{aligned} $$
We can set $J_{\mathbf{R}}^{*j}(\theta, -N)$ such that

$$
\begin{align*}
J_{\mathbf{R}}^{*j}(\theta, -N) = & (-1)^N N! \sum_{\mathcal{P} \subseteq [1, P]} \prod_{p \in \mathcal{P}} c_{j,p}^{-R_p-1} R_p! \sum_{\substack{k=(k_p)_{p \in \mathcal{P}^c} \in \mathbb{N}_0^{\mathcal{P}^c} \\ |k| \neq N + |\mathbf{R}_{||p||} + |\mathcal{P}|}} (-1)^{|k|} \left( \prod_{p \in \mathcal{P}^c} c_{j,p}^{k_p} \frac{\zeta(-R_p - k_p, d_p)}{k_p!} \right) \frac{\theta^{-N-|\mathbf{R}_{||P||}-|\mathcal{P}||+|k||}}{-N - |\mathbf{R}_{||P||} - |\mathcal{P}|| + |k||} \\
& + N! \sum_{\mathcal{P} \subseteq [1, P]} (-1)^{|\mathbf{R}_{||P||} + |\mathcal{P}||} \prod_{p \in \mathcal{P}} c_{j,p}^{-R_p-1} R_p! \sum_{\substack{k=(k_p)_{p \in \mathcal{P}^c} \in \mathbb{N}_0^{\mathcal{P}^c} \\ |k|=N+|\mathbf{R}_{||P||} + |\mathcal{P}||}} \left( \prod_{p \in \mathcal{P}^c} c_{j,p}^{k_p} \frac{\zeta(-R_p - k_p, d_p)}{k_p!} \right) \ln(\theta) \\
& + (-1)^N N! \left( \prod_{p=1}^P R_p! \right) \left( \prod_{p=1}^P c_{j,p}^{-R_p-1} \right) \frac{\theta^{-N-|\mathbf{R}|-P}}{-N - |\mathbf{R}|-P}.
\end{align*}
$$

□

**Remark 5.2.3.** The term $J_{\mathbf{R}}^{*j}(\theta, -N)$ corresponds to a sum of a multiple of $\ln(\theta)$ and a Laurent series without any constant term.

Let us observe that we cannot obtain an explicit expression for the values of $\partial_s \varphi_{\mathbf{R}}^j$ at nonpositive integers using (5.4) and (5.5), since it is not true that the term $\partial_s K_{\mathbf{R}}^j(\theta, -N)$ vanishes. We thus need another strategy to compute the values of $\partial_s \varphi_{\mathbf{R}}^j$ at nonpositive integers.

5.2.1 Proof of Proposition B

In this section, we aim to provide explicit values of the derivatives of these auxiliary functions assuming
the coefficients $(c_{q,p})_{q \in [1,Q] \times [1,P]}$ are rationals.

Let $P \in \mathbb{N}$, and $\mathbf{R} \in \mathbb{N}_0^P$. To simplify notations in this subsection, we will write for all complex $s$ such that $\sigma \gg_{\mathbf{R}} 1$

$$
\varphi_{\mathbf{R}}(s, \mathbf{d}|w_1, \dots, w_P) := \sum_{\mathbf{n} \in \mathbb{N}_0^P} \frac{(n_1 + d_1)^{R_1} \dots (n_P + d_P)^{R_P}}{\left(\sum_{p=1}^P w_p (n_p + d_p)\right)^s},
$$

with **d** = (d_p)_{p∈[1,P]} and w_p, d_p ∈ H_0 for all integer p. This function admits a meromorphic extension to ℂ in the variable s. In this subsection, we wish to establish an explicit relation between ϕ_R^j(s) and the Hurwitz zeta function ζ(s, x), with x ∈ H_0. To do this, we follow a strategy developed by Aoki and Sakane in [SA22], which was initially used to get formulas for special values of the higher derivatives of Barnes zeta functions with rational coefficients. Here, the auxiliary functions ϕ_R(s, x|w_1, ..., w_P) are not exactly Barnes zeta functions due to the terms in the numerator of the series, however, the strategy used by Aoki and Sakane works in the same way to study this function.

**Lemma 5.2.4.** Let $w_1, \dots, w_P \in \mathbb{Q}_+^*$ such that $w_p = \frac{a_p}{b_p}$ with $a_p$ and $b_p$ are positive integers. For any common multiple $\alpha$ of $b_1, \dots, b_P$, and any common multiple $\beta$ of $\alpha w_1, \dots, \alpha w_P$, we have

$$
\varphi_{\mathbf{R}}(s, \mathbf{d}|w_1, \dots, w_P) = \left(\frac{\alpha}{\beta}\right)^s \prod_{p=1}^{P} \beta_p^{R_p} \sum_{\substack{0 \le k_1 \le \beta_1 - 1 \\ 0 \le k_P \le \beta_P - 1}} \varphi_{\mathbf{R}} \left( s, \left( \frac{d_p + k_p}{\beta_p} \right)_{p \in [1, P]} |1\rangle \right), \quad (5.6)
$$

$$
\text{with } \beta_1 = \frac{\beta}{\alpha w_1}, \dots, \beta_P = \frac{\beta}{\alpha w_P}.
$$
**Proof.** Let $s \in \mathbb{C}$ such that $\sigma \gg 1$. In order to get the required result, we will decompose each sum $\sum_{n_p=0}^{+\infty}$

using $\sum_{n_p=0}^{+\infty}$, with $k_p$ running over the set $[0, \beta_p - 1]$. We then get

$$ \varphi_{\mathbf{R}}(s, \mathbf{d}|w_1, \dots, w_P) = \left(\frac{\alpha}{\beta}\right)^s \sum_{k_1=0}^{\beta_1-1} \dots \sum_{k_P=0}^{\beta_P-1} \sum_{\substack{n \in \mathbb{N}_0^P \\ \forall p, n_p = k_p \bmod \beta_p}} \frac{\prod_{p=1}^P (n_p + d_p)^{R_p}}{\left(\frac{\alpha}{\beta} \left(\sum_{p=1}^P (w_p d_p + w_p k_p + w_p (n_p - k_p))\right)\right)^s}. $$

By a change of variables $n'_p = \beta_p n_p + k_p$ for each integer $p$, we get

$$ \varphi_{\mathbf{R}}(s, \mathbf{d}|w_1, \dots, w_P) = \left(\frac{\alpha}{\beta}\right)^s \sum_{k_1=0}^{\beta_1-1} \dots \sum_{k_P=0}^{\beta_P-1} \sum_{\mathbf{n} \in \mathbb{N}_0^P} \frac{\prod_{p=1}^P (\beta_p n_p + k_p + d_p)^{R_p}}{\left(\frac{\alpha}{\beta} \sum_{p=1}^P w_p d_p + w_p k_p + w_p \beta_p n_p\right)^s}. $$

By factoring $\beta_p^{R_p}$ in the numerator, and by noticing that $\frac{\alpha w_p \beta_p}{\beta} = 1$, we get

$$ \varphi_{\mathbf{R}}(s, \mathbf{d}|w_1, \dots, w_P) = \left(\frac{\alpha}{\beta}\right)^s \prod_{p=1}^{P} \beta_p^{R_p} \sum_{k_1=0}^{\beta_1-1} \dots \sum_{k_P=0}^{\beta_P-1} \sum_{\mathbf{n} \in \mathbb{N}_0^P} \frac{\prod_{p=1}^{P} \left(n_p + \frac{k_p+d_p}{\beta_p}\right)^{R_p}}{\left(\sum_{p=1}^{P} n_p + \frac{\alpha}{\beta} (w_p d_p + w_p k_p)\right)^s}. $$

Finally, one can check that

$$ \frac{k_p + d_p}{\beta_p} = \frac{\alpha}{\beta} (w_p d_p + w_p k_p), $$

for all $1 \le p \le P$. We then find that the formula 5.6 is true for all $s \in \mathbb{C}$ such that $\sigma \gg 1$. We can then verify that the formula is true on all $\mathbb{C}$ by analytic continuation. $\square$

We can specify the last formula 5.6 by taking an explicit rational coefficient $\frac{\alpha}{\beta}$. In fact, we can directly consider a coefficient computed through the $\text{lcm}$ and the $\text{gcd}$ of the numerator and denominator of $w_1, \dots, w_P$, which allows us for more explicit results later on.

**Proposition 5.2.5.** Let $\mathbf{d} \in H_0$ $w_1 = \frac{a_1}{b_1}, \dots, w_P = \frac{a_P}{b_P}$ be rational numbers with $a_p, b_p > 0$ coprime integers for all $p \in [1, P]$. We set $w := \frac{\text{lcm}(a_1, \dots, a_P)}{\text{gcd}(b_1, \dots, b_P)}$, and for all $p \in [1, P]$, we define $\beta_1 := \frac{w}{w_1}, \dots, \beta_P := \frac{w}{w_P}$. We then have

$$ \varphi_{\mathbf{R}}(s, \mathbf{d}|w_1, \dots, w_P) = w^{-s} \prod_{p=1}^{P} \beta_p^{R_p} \sum_{\substack{0 \le k_1 \le \beta_1 - 1 \\ 0 \le k_P \le \beta_P - 1}} \varphi_{\mathbf{R}} \left( s, \left( \frac{d_p + k_p}{\beta_p} \right)_{p \in [1, P]} |\mathbf{1} \right). $$

**Proof.** Let $\alpha = \text{lcm}(b_1, \dots, b_P)$ and $\beta = \text{lcm}(\alpha w_1, \dots, \alpha w_P)$. By [SA22, p.3], we know that $\frac{\alpha}{\beta} = w^{-1}$. Therefore, we can use formula 5.6 with such $\alpha$ and $\beta$. $\square$

Finally, we obtain a formula similar to the one proved in [Ono21, Proposition 4.1] to re-express $\varphi_{\mathbf{R}}(s, \mathbf{d}|1, \dots, 1)$, by making a very slight modification to Onodera's proof.

**Proposition 5.2.6.** Let $\mathbf{d} \in H_0^P$, $\mathbf{R} \in \mathbb{N}_0^P$, and $s \in \mathbb{C} \setminus \mathbb{N}$. We then have that

$$ \varphi_{\mathbf{R}}(s, \mathbf{d}|\mathbf{1}) = \sum_{\mathcal{P} \subseteq [1, P]} (-1)^{|\mathcal{P}|} \prod_{p \in \mathcal{P}} R_p! \sum_{\substack{\mathbf{k} \in \mathbb{N}_0^P, k' \ge 0 \\ |\mathbf{k}|_{\mathcal{P}} + k' = |\mathbf{R}|_{\mathcal{P}} + |\mathcal{P}| - 1}} (-1)^{|\mathbf{k}|} \frac{\zeta(s - k', |\mathbf{d}|)}{k'!} \prod_{p \in \mathcal{P}} \frac{\zeta(-R_p - k_p, d_p)}{k_p!}. $$
In order to prove such formula, we need to use a lemma proved by Onodera [Ono21, Lemma 4.2], which generalizes Faulhaber's formula:

$$ \sum_{n=1}^{y} n^r = \frac{1}{r+1} \sum_{k=0}^{r} \binom{r+1}{k} (-1)^k B_k y^{r-k+1}. $$

**Lemma 5.2.7.** [Ono21] Let $\mathbf{R} = (R_1, ..., R_P) \in \mathbb{N}_0^P$ be nonnegative integers, and $y \in \mathbb{Z}$ an integer. We have that

$$ \sum_{\substack{n \in \mathbb{N}_0^P \\ |\mathbf{n}|=y}} \prod_{p=1}^{P} (n_p + d_p)^{R_p} + (-1)^{P-1} \sum_{\substack{n \in \mathbb{Z}^P \\ |\mathbf{n}|=-1}} \prod_{p=1}^{P} (n_p + d_p)^{R_p} = \\ \sum_{\mathcal{P} \subseteq [1, P]} \prod_{p \in \mathcal{P}^c} R_p! \sum_{\substack{k \in \mathbb{N}_0^P, k' \in \mathbb{N}_0 \\ |\mathbf{k}|+k'=|\mathbf{R}|+|\mathcal{P}^c|+|\mathcal{P}^c|-1}} \frac{(|\mathbf{d}|+y)^{k'}}{k'!} \prod_{p \in \mathcal{P}} (-1)^{k_p} \frac{\zeta(-R_p - k_p, d_p)}{k_p!}. $$

Let's note that if $y \in \mathbb{N}_0$, then the second sum in the last equality vanishes. We can now prove Proposition 5.2.6:

*Proof.* Let's first note that, for all $s \in \mathbb{C}$ such that $\sigma \gg 1$, we have

$$ \varphi_{\mathbf{R}}(s, \mathbf{d}|\mathbf{1}) = \sum_{n'=0}^{+\infty} \frac{1}{(n'+|\mathbf{d}|)^s} \sum_{\substack{n \in \mathbb{N}_0^P \\ |\mathbf{n}|=n'}} \prod_{p=1}^{P} (n_p + d_p)^{R_p}. $$

By Lemma 5.2.7, we then get that

$$ \begin{aligned} \varphi_{\mathbf{R}}(s, \mathbf{d}|\mathbf{1}) &= \sum_{\mathcal{P} \subseteq [1, P]} \prod_{p \in \mathcal{P}^c} R_p! \sum_{\substack{k \in \mathbb{N}_0^P, k' \in \mathbb{N}_0 \\ |\mathbf{k}|+k'=|\mathbf{R}|+|\mathcal{P}^c|+|\mathcal{P}^c|-1}} \prod_{p \in \mathcal{P}} (-1)^{k_p} \frac{\zeta(-R_p - k_p, d_p)}{k_p!} \sum_{n'=0}^{+\infty} \frac{(n'+|\mathbf{d}|)^{k'}}{k'!} \frac{1}{(n'+|\mathbf{d}|)^s} \\ &= \sum_{\mathcal{P} \subseteq [1, P]} \prod_{p \in \mathcal{P}^c} R_p! \sum_{\substack{k \in \mathbb{N}_0^P, k' \in \mathbb{N}_0 \\ |\mathbf{k}|+k'=|\mathbf{R}|+|\mathcal{P}^c|+|\mathcal{P}^c|-1}} (-1)^{|\mathbf{k}|} \left( \prod_{p \in \mathcal{P}} \frac{\zeta(-R_p - k_p, d_p)}{k_p!} \right) \frac{\zeta(s - k', |\mathbf{d}|)}{k'!}. \end{aligned} $$

Applying Proposition 5.2.6 into the formula obtained in Proposition 5.2.5, we get the following proposition:

**Proposition 5.2.8.** Let $\mathbf{d} \in H_0^P$, $\mathbf{R} \in \mathbb{N}_0^P$, $s \in \mathbb{C} \setminus \mathbb{N}$, and $w_1 = \frac{a_1}{b_1}, ..., w_P = \frac{a_P}{b_P}$ be rational numbers with $a_p, b_p > 0$ coprime integers for all $p \in [1, P]$. We define $w := \frac{\text{lcm}(a_1, ..., a_P)}{\text{gcd}(b_1, ..., b_P)}$, and for all $p \in [1, P]$, we define $\beta_1 := \frac{w}{w_1}, ..., \beta_P := \frac{w}{w_P}$. Then we have

$$ \varphi_{\mathbf{R}}(s, \mathbf{d}|w_1, ..., w_P) = w^{-s} \prod_{p=1}^{P} \beta_p^{R_p} \sum_{\substack{0 \le u_1 \le \beta_1 - 1 \\ \vdots \\ 0 \le u_P \le \beta_P - 1}} \sum_{\mathcal{P} \subseteq [1, P]} \prod_{p \in \mathcal{P}^c} R_p! \sum_{\substack{k \in \mathbb{N}_0^P, k' \in \mathbb{N}_0 \\ |\mathbf{k}|+k'=|\mathbf{R}|+|\mathcal{P}^c|+|\mathcal{P}^c|-1}} (-1)^{|\mathbf{k}|} \prod_{p \in \mathcal{P}} \frac{\zeta\left(-R_p - k_p, \frac{d_p+u_p}{\beta_p}\right)}{k_p!} \cdot \frac{\zeta(s - k', w^{-1}(w_1(d_1+u_1) + \dots + w_P(d_P+u_P)))}{k'!} $$
## 5.3 Proof of Theorem D

Let $(\mathbf{N}, \mathbf{N}') \in \mathbb{N}_0^P \times \mathbb{N}_0^Q, (\boldsymbol{\mu}, \boldsymbol{\mu}') \in \mathbb{R}_+^P \times \mathbb{R}_+^Q$. By Proposition 2.2.4, for all complex $s$ around 0, and for all $0 < \theta \ll 1$ we have that

$$Z_{\mathbf{N},\mathbf{N}'}(s) = K_{\mathbf{N},\mathbf{N}'}(\theta, s) + J_{\mathbf{N},\mathbf{N}'}(\theta, s),$$

with

$$K_{\mathbf{N},\mathbf{N}'}(\theta, s) = \sum_{\substack{Q \subset [1,Q] \\ Q \neq \emptyset}} \sum_{A_Q \subset Q^c} (-1)^{|Q^c| - |A_Q|} \frac{1}{\prod_{q \in [1,Q] \setminus A_Q} \Gamma(-N_q' + \mu_q's)} \left( \sum_{\mathbf{n} \in \mathbb{N}_0^P} \prod_{p=1}^P (n_p + d_p)^{N_p - \mu_p s} \cdot \prod_{q \in A_Q} (l_q(\mathbf{n}) + d_q')^{N_q' - \mu_q's} \prod_{q \in [1,Q] \setminus A_Q} \Gamma(-N_q' + \mu_q's, \theta, l_q(\mathbf{n}) + d_q') \right),$$

and

$$J_{\mathbf{N},\mathbf{N}'}(\theta, s) = \sum_{\mathcal{P} \subseteq [1,P]} \sum_{j=1}^{Q} \prod_{p \in \mathcal{P}} \Gamma(1 + N_p - \mu_p s) \cdot \sum_{\mathbf{k}=(k_p)_{p \in \mathcal{P}} \in \mathbb{N}_0^{P^c}} (-1)^{|\mathbf{k}|} h_{\mathbf{N},\mathbf{N}',\mathcal{P},j,\mathbf{k}}(s) \frac{\theta^{|\mathbf{\mu}'|+|\mathbf{\mu}|_\mathcal{P}} s - |\mathbf{N}'| - |\mathbf{N}|_\mathcal{P} - |\mathcal{P}| + |\mathbf{k}|_\mathcal{P}^c}{\Gamma(-N_j' + \mu_j's)((|\mathbf{\mu}'| + |\mathbf{\mu}|_\mathcal{P})s - |\mathbf{N}'| - |\mathbf{N}|_\mathcal{P} - |\mathcal{P}| + |\mathbf{k}|_\mathcal{P}^c)} \cdot \prod_{p \in \mathcal{P}^c} \frac{\zeta(-N_p + \mu_p s - k_p, d_p)}{k_p!} + \left( \prod_{p=1}^P \Gamma(1 + N_p - \mu_p s) \right) \left( \sum_{j=1}^Q h_{\mathbf{N},\mathbf{N}',[1,P],j}(s) \frac{\theta^{|\mathbf{\mu}'|+|\mathbf{\mu}|} s - |\mathbf{N}'| - |\mathbf{N}| - P}{\Gamma(-N_j' + \mu_j's)((|\mathbf{\mu}'| + |\mathbf{\mu}|)s - |\mathbf{N}'| - |\mathbf{N}| - P)} \right).$$

**Proposition 5.3.1.** Let $(\mathbf{N}, \mathbf{N}') \in \mathbb{N}_0^P \times \mathbb{N}_0^Q, (\boldsymbol{\mu}, \boldsymbol{\mu}') \in \mathbb{R}_+^P \times \mathbb{R}_+^Q$. There exist a function $K_{\mathbf{N},\mathbf{N}'}^*(\theta)$ defined for all $0 < \theta \ll 1$, such that

$$\forall 0 < \theta \ll 1, \quad K_{\mathbf{N},\mathbf{N}'}^*(\theta) = \beta \ln(\theta) + \sum_{n=m}^{+\infty} \alpha_n \theta^n,$$

with $m \in \mathbb{Z}$ and integer, $\beta \in \mathbb{C}$ a complex number, and $(\alpha_n)_{n \in [m, +\infty[}$ a complex sequence such that
$\alpha_0 = 0$, and such that:

$$ \partial_s K_{\mathbf{N},\mathbf{N}'}(\theta, s)_{s=0} = \tag{5.7} $$

$$
\begin{aligned}
& \sum_{j=1}^{Q} \mu_j' \quad \sum_{\substack{\mathbf{u}=(\mathbf{u}_q)_{q \in [1,Q]} \setminus \{j\} \in (\mathbb{N}_0^P)^{[1,Q]} \setminus \{j\} \\ \forall q \in [1,Q] \setminus \{j\}, |\mathbf{u}_q|=N'_q}} \prod_{\substack{q=1 \\ q \neq j}}^{Q} \binom{N'_q}{\mathbf{u}_q} \prod_{p=1}^{P} c_{q,p}^{\mathbf{u}(p)} \partial_s \varphi^j_{(N_p+\mathbf{u}(p))_{p \in [1,P]}} (s)|_{s=-N'_j} \\
& - \sum_{j=1}^{Q} (\gamma - h_{N'_j}) \mu'_j N'_j! \quad \sum_{\substack{\mathbf{u}=(\mathbf{u}_q)_{q \in [1,Q]} \setminus \{j\} \in (\mathbb{N}_0^P)^{[1,Q]} \setminus \{j\} \\ \forall q \in [1,Q] \setminus \{j\}, |\mathbf{u}_q|=N'_q}} \left( \left( \prod_{\substack{q=1 \\ q \neq j}}^{Q} \binom{N'_q}{\mathbf{u}_q} \prod_{p=1}^{P} c_{q,p}^{\mathbf{u}(p)} \right) \right. \\
& \qquad \left. \cdot \sum_{\mathcal{P} \subseteq [1,P]} (-1)^{|\mathbf{R}(\mathbf{N},j,\mathbf{u})|_{\mathcal{P}} + |\mathcal{P}|} \prod_{p \in \mathcal{P}} c_{j,p}^{-R(\mathbf{N},j,\mathbf{u})_p - 1} R(\mathbf{N},j,\mathbf{u})_p! \quad \sum_{\substack{\mathbf{k}=(k_p) \in \mathcal{P}^c \in \mathbb{N}_0^{P^c} \\ |\mathbf{k}|=N'_j + |\mathbf{R}(\mathbf{N},j,\mathbf{u})|_{|\mathcal{P}|+|\mathcal{P}|}}} \prod_{p \in \mathcal{P}^c} c_{j,p}^{k_p} \frac{\zeta(-\mathbf{R}(\mathbf{N},j,\mathbf{u})_p - k_p, d_p)}{k_p!} \right)
\end{aligned}
$$

*Proof.* Taking the derivative of $K_{\mathbf{N},\mathbf{N}'}(\theta, s)$ with respect to $s$ in formula 2.4, we get

$$ \partial_s K_{\mathbf{N},\mathbf{N}'}(\theta, s) = \tag{5.8} $$

$$
\begin{aligned}
& \sum_{\substack{\mathcal{Q} \subset [1,Q] \\ \mathcal{Q} \neq \emptyset}} \sum_{A_{\mathcal{Q}} \subset \mathcal{Q}^c} (-1)^{|\mathcal{Q}^c| - |A_{\mathcal{Q}}|} \partial_s \left( \prod_{q \in [1,Q] \setminus A_{\mathcal{Q}}} \frac{1}{\Gamma(-N'_q + \mu'_q s)} \right) \left( \sum_{\mathbf{n} \in \mathbb{N}_0^P} \prod_{p=1}^P (n_p + d_p)^{N_p - \mu_p s} \right. \\
& \qquad \left. \cdot \prod_{q \in A_{\mathcal{Q}}} (l_q(\mathbf{n}) + d'_q)^{N'_q - \mu'_q s} \prod_{q \in [1,Q] \setminus A_{\mathcal{Q}}} \Gamma(-N'_q + \mu'_q s, \theta, l_q(\mathbf{n}) + d'_q) \right) \\
& + \sum_{\substack{\mathcal{Q} \subset [1,Q] \\ \mathcal{Q} \neq \emptyset}} \sum_{A_{\mathcal{Q}} \subset \mathcal{Q}^c} (-1)^{|\mathcal{Q}^c| - |A_{\mathcal{Q}}|} \prod_{q \in [1,Q] \setminus A_{\mathcal{Q}}} \frac{1}{\Gamma(-N'_q + \mu'_q s)} \partial_s \left( \sum_{\mathbf{n} \in \mathbb{N}_0^P} \prod_{p=1}^P (n_p + d_p)^{N_p - \mu_p s} \right. \\
& \qquad \left. \cdot \prod_{q \in A_{\mathcal{Q}}} (l_q(\mathbf{n}) + d'_q)^{N'_q - \mu'_q s} \prod_{q \in [1,Q] \setminus A_{\mathcal{Q}}} \Gamma(-N'_q + \mu'_q s, \theta, l_q(\mathbf{n}) + d'_q) \right).
\end{aligned}
$$

Thanks to the bound obtained in 2.12, we've already seen in the proof of Proposition 2.3.11 that the function

$$ s \in \mathbb{C} \mapsto \left( \sum_{\mathbf{n} \in \mathbb{N}_0^P} \prod_{p=1}^P (n_p + d_p)^{N_p - \mu_p s} \prod_{q \in A_{\mathcal{Q}}} (l_q(\mathbf{n}) + d'_q)^{N'_q - \mu'_q s} \prod_{q \in [1,Q] \setminus A_{\mathcal{Q}}} \Gamma(-N'_q + \mu'_q s, \theta, l_q(\mathbf{n}) + d'_q) \right) $$

was holomorphic on $\mathbb{C}$. As $s \mapsto \frac{1}{\Gamma(s)}$ vanish at nonpositive integers, we find that the second term in 5.8 vanish at $s = 0$. Moreover, we know that around $s = 0$, we have

$$ \frac{1}{\Gamma(-N'_{q} + \mu'_{q}s)} = O(s). $$
Therefore, we find that for all $\emptyset \neq Q \subset [1, Q]$ and for all $A_Q \subset Q^c = [1, Q] \setminus Q$, we have

$$ \prod_{q \in [1, Q] \setminus A_Q} \frac{1}{\Gamma(-N'_q + \mu'_q s)} = \begin{cases} O(s^2) & \text{if } |Q| \ge 2, \\ O(s) & \text{if } Q = \{j\}, A_Q = [1, Q] \setminus \{j\} \text{ with } j \in [1, Q], \\ O(s^2) & \text{otherwise.} \end{cases} $$

Therefore, we obtain that if $|Q| \ge 2$, we have

$$ \partial_s \left( \prod_{q \in [1, Q] \setminus A_Q} \frac{1}{\Gamma(-N'_q + \mu'_q s)} \right)_{s=0} = 0. $$

We then get

$$ \partial_s K_{\mathbf{N}, \mathbf{N}'}(\theta, s)_{s=0} = \sum_{j=1}^{Q} \partial_s \left( \frac{1}{\Gamma(-N'_j + \mu'_j s)} \right)_{s=0} \left( \sum_{\mathbf{n} \in \mathbb{N}_0^P} \Gamma(-N'_j, \theta, l_j(\mathbf{n}) + d'_j) \prod_{p=1}^{P} (n_p + d_p)^{N_p} \cdot \prod_{\substack{q=1 \\ q \neq j}}^{Q} (l_q(\mathbf{n}) + d'_q)^{N'_q} \right). $$

By Lemma 3.1.1, we obtain

$$ \partial_s \left( \frac{1}{\Gamma(-N'_j + \mu'_j s)} \right)_{s=0} = (-1)^{N'_j} N'_j! \mu'_j, $$

and therefore

$$ \partial_s K_{\mathbf{N}, \mathbf{N}'}(\theta, s)_{s=0} = \sum_{j=1}^{Q} \mu'_j (-1)^{N'_j} N'_j! \left( \sum_{\mathbf{n} \in \mathbb{N}_0^P} \Gamma(-N'_j, \theta, l_j(\mathbf{n}) + d'_j) \prod_{p=1}^{P} (n_p + d_p)^{N_p} \cdot \prod_{\substack{q=1 \\ q \neq j}}^{Q} (l_q(\mathbf{n}) + d'_q)^{N'_q} \right). $$

For all $q \in [1, Q]$, $\mathbf{n} \in \mathbb{N}_0^P$, we have $l_q(\mathbf{n}) + d'_q = \sum_{p=1}^P c_{q,p}(n_p + d_p)$. By Newton's multinomial, we get

$$ (l_q(\mathbf{n}) + d'_q)^{N'_q} = \sum_{\substack{\mathbf{u_q} \in \mathbb{N}_0^P \\ |\mathbf{u_q}| = N'_q}} \left( \frac{N'_q}{\mathbf{u_q}} \right) \prod_{p=1}^{P} c_{q,p}^{u_{q,p}} (n_p + d_p)^{u_{q,p}}. $$

By taking the product over $q \in [1, Q] \setminus \{j\}$, we find

$$ \begin{aligned} \prod_{\substack{q=1 \\ q \neq j}}^{Q} (l_q(\mathbf{n}) + d'_q)^{N'_q} &= \sum_{\substack{\mathbf{u}=(\mathbf{u}_q)_{q \in [1,Q] \setminus \{j\}} \in (\mathbb{N}_0^P)^{[1,Q] \setminus \{j\}} \\ \forall q \in [1,Q] \setminus \{j\}, |\mathbf{u}_q|=N'_q}} \prod_{\substack{q=1 \\ q \neq j}}^{Q} \left( \left( \binom{N'_q}{\mathbf{u}_q} \right) \prod_{p=1}^{P} c_{q,p}^{u_{q,p}} (n_p + d_p)^{u_{q,p}} \right) \\ &= \sum_{\substack{\mathbf{u}=(\mathbf{u}_q)_{q \in [1,Q] \setminus \{j\}} \in (\mathbb{N}_0^P)^{[1,Q] \setminus \{j\}} \\ \forall q \in [1,Q] \setminus \{j\}, |\mathbf{u}_q|=N'_q}} \left( \prod_{\substack{q=1 \\ q \neq j}}^{Q} \left( \binom{N'_q}{\mathbf{u}_q} \right) \prod_{p=1}^{P} c_{q,p}^{u_{q,p}} \right) \prod_{p=1}^{P} (n_p + d_p)^{\mathbf{u}(p)}, \end{aligned} $$
where we noted

$$ \mathbf{u}(p) = \sum_{\substack{q=1 \\ q \neq j}}^{Q} u_{q,p}. $$

We finally get

$$ \partial_s K_{\mathbf{N},\mathbf{N}'}(\theta, s)_{s=0} = \sum_{j \in [1,Q]} \mu_j' (-1)^{N_j'} N_j'! \sum_{\substack{\mathbf{u}=(\mathbf{u}_q)_{q \in [1,Q]\setminus\{j\}} \in (\mathbb{N}_0^P)^{[1,Q]\setminus\{j\}} \\ \forall q \in [1,Q]\setminus\{j\}, |\mathbf{u}_q|=N_q'} \prod_{q=1}^{Q} \binom{N_q'}{q \neq j} \prod_{p=1}^{P} c_{q,p}^{u_{q,p}} \left( \sum_{\mathbf{n} \in \mathbb{N}_0^P} \Gamma(-N_j', \theta, l_j(\mathbf{n}) + d_j') \prod_{p=1}^{P} (n_p + d_p)^{N_p+\mathbf{u}(p)} \right). \quad (5.9) $$

By formula (5.2) we know that for all $m \in \mathbb{N}_0$,

$$ (-1)^m m! \sum_{\mathbf{n} \in \mathbb{N}_0^P} \Gamma(-m, \theta, l_j(\mathbf{n}) + d_j') \prod_{p=1}^P (n_p + d_p)^{R_p} = \partial_s \varphi_{\mathbf{R}}^j(s)|_{s=-m} - \partial_s J_{\mathbf{R}}^j(\theta, s)|_{s=-m}. $$

Substituting $(-1)^m m! \sum_{\mathbf{n} \in \mathbb{N}_0^P} \Gamma(-m, \theta, l_j(\mathbf{n}) + d_j') \prod_{p=1}^P (n_p + d_p)^{R_p}$ in 5.9 by the above formula, we get

$$ \partial_s K_{\mathbf{N},\mathbf{N}'}(\theta, s)_{s=0} = \sum_{j=1}^{Q} \mu_j' \sum_{\substack{\mathbf{u}=(\mathbf{u}_q)_{q \in [1,Q]\setminus\{j\}} \in (\mathbb{N}_0^P)^{[1,Q]\setminus\{j\}} \\ \forall q \in [1,Q]\setminus\{j\}, |\mathbf{u}_q|=N_q'}} \prod_{q=1}^{Q} \binom{N_q'}{q \neq j} \prod_{p=1}^{P} c_{q,p}^{u_{q,p}} \quad (5.10) \\ \left( \partial_s \varphi_{(N_p+\mathbf{u}(p))_{p \in [1,P]}}^j(s) \right)_{s=-N_j'} - \partial_s J_{(N_p+\mathbf{u}(p))_{p \in [1,P]}}^j(\theta, s) \left. \right|_{s=-N_j'} $$

Let's note $\mathbf{R}(\mathbf{N}, j, \mathbf{u}) = (N_p + \mathbf{u}(p))_{p \in [1, P]}$ for all $\mathbf{u} = (\mathbf{u}_q)_{q \in [1, Q] \setminus \{j\}}$. By formula (5.5), we get

$$ \partial_s J_{\mathbf{R}(\mathbf{N},j,\mathbf{u})}^j(\theta, s)_{s=-N_j'} = (\gamma - h_{N_j'}) N_j'! \sum_{\mathcal{P} \subset [1,P]} (-1)^{|\mathbf{R}(\mathbf{N},j,\mathbf{u})|_{\mathcal{P}} + |\mathcal{P}|} \prod_{p \in \mathcal{P}} c_{j,p}^{-R(\mathbf{N},j,\mathbf{u})_p - 1} R(\mathbf{N},j,\mathbf{u})_p! \\ \sum_{\substack{\mathbf{k}=(k_p)_{p \in \mathcal{P}} \in \mathbb{N}_0^{P_c} \\ |\mathbf{k}|=N_j' + |\mathbf{R}(\mathbf{N},j,\mathbf{u})|_{\mathcal{P}} + |\mathcal{P}|}} \prod_{p \in \mathcal{P}} c_{j,p}^{k_p} \frac{\zeta(-\mathbf{R}(\mathbf{N},j,\mathbf{u})_p - k_p, d_p)}{k_p!} \quad (5.11) \\ + J_{\mathbf{R}(\mathbf{N},j,\mathbf{u})}^{*j}(\theta, -N_j'). $$

For all $0 < \theta \ll 1$, we set

$$ K_{\mathbf{N},\mathbf{N}'}^{*}(\theta) := -\sum_{j=1}^{Q} \mu_j' \sum_{\substack{\mathbf{u}=(\mathbf{u}_q)_{q \in [1,Q]\setminus\{j\}} \in (\mathbb{N}_0^P)^{[1,Q]\setminus\{j\}} \\ \forall q \in [1,Q]\setminus\{j\}, |\mathbf{u}_q|=N_q'}} J_{\mathbf{R}(\mathbf{N},j,\mathbf{u})}^{*j}(\theta, -N_j') \prod_{q=1}^{Q} \binom{N_q'}{q \neq j} \prod_{p=1}^{P} c_{q,p}^{u_{q,p}}. $$

By definition of $J_{\mathbf{R}(\mathbf{N},j,\mathbf{u})}^{*j}(\theta, -N_j')$, we see that $K_{\mathbf{N},\mathbf{N}'}^{*}(\theta)$ are of the forms $\sum_{n \ge m} \alpha_n \theta^n + \beta \ln(\theta)$ with $\alpha_n$ and $\beta$ complex numbers for all $n \ge m$, and $\alpha_0 = 0$. Injecting formula 5.11 into formula 5.10, we find the
required formula:

$$ \partial_s K_{\mathbf{N},\mathbf{N}'}(\theta, s)_{s=0} = \tag{5.12} $$

$$
\begin{aligned}
& \sum_{j=1}^{Q} \mu'_{j} \quad \sum_{\substack{\mathbf{u}=(\mathbf{u}_{\mathbf{q}})_{q \in [1,Q]} \setminus \{j\} \in (\mathbb{N}_0^P)^{[1,Q]} \setminus \{j\} \\ \forall q \in [1,Q] \setminus \{j\}, |\mathbf{u}_{\mathbf{q}}|=N'_q}} \prod_{\substack{q=1 \\ q \neq j}}^{Q} \binom{N'_q}{\mathbf{u}_{\mathbf{q}}} \prod_{p=1}^{P} c_{q,p}^{u_{q,p}} \partial_s \varphi^{j}_{(N_p+\mathbf{u}(p))_{p \in [1,P]}} (s)|_{s=-N'_j} \\
& - \sum_{j=1}^{Q} \mu'_{j} \quad \sum_{\substack{\mathbf{u}=(\mathbf{u}_{\mathbf{q}})_{q \in [1,Q]} \setminus \{j\} \in (\mathbb{N}_0^P)^{[1,Q]} \setminus \{j\} \\ \forall q \in [1,Q] \setminus \{j\}, |\mathbf{u}_{\mathbf{q}}|=N'_q}} \left( \left( \prod_{\substack{q=1 \\ q \neq j}}^{Q} \binom{N'_q}{\mathbf{u}_{\mathbf{q}}} \right) \prod_{p=1}^{P} c_{q,p}^{u_{q,p}} (-1)^{N'_j} N'_j! (\gamma - h_{N'_j}) \right) \\
& \cdot \sum_{\mathcal{P} \subseteq [1,P]} (-1)^{|\mathbf{R}(\mathbf{N},j,\mathbf{u})|_{||\mathcal{P}|+||\mathcal{P}|}} \prod_{p \in \mathcal{P}} c_{j,p}^{-R(\mathbf{N},j,\mathbf{u})_p - 1} R(\mathbf{N},j,\mathbf{u})_p! \quad \sum_{\substack{\mathbf{k}=(k_p)_{p \in \mathcal{P}^c} \in \mathbb{N}_0^{\mathcal{P}^c} \\ |\mathbf{k}|=N'_j + |\mathbf{R}(\mathbf{N},j,\mathbf{u})|_{||\mathcal{P}|+||\mathcal{P}|}}} \prod_{p \in \mathcal{P}^c} c_{j,p}^{k_p} \frac{\zeta(-\mathbf{R}(\mathbf{N},j,\mathbf{u})_p - k_p, d_p)}{k_p!} \\
& + K_{\mathbf{N},\mathbf{N}'}^*(\theta)_{\mu,\mu'}
\end{aligned}
$$

<figure><img src="image_19.png" /></figure>

We then just need to compute $\partial_s(J_{\mathbf{N},\mathbf{N}'}(\theta, s))$ at $s = 0$ in order to prove Theorem D.

**Proposition 5.3.2.** Let $(\mathbf{N}, \mathbf{N}') \in \mathbb{N}_0^P \times \mathbb{N}_0^Q$, and $(\mu, \mu') \in \mathbb{R}^{+P} \times \mathbb{R}_*^{+Q}$. There exist a function $J_{\mathbf{N},\mathbf{N}'}^*(\theta)$
defined for all $0 < \theta \ll 1$, such that

$$ \forall 0 < \theta \ll 1, \quad J_{\mathbf{N},\mathbf{N}'}^*(\theta) = \beta \ln(\theta) + \sum_{n=m}^{+\infty} \alpha_n \theta^n, $$

with $m \in \mathbb{Z}$ an integer, $\beta \in \mathbb{R}$ a complex number and $(\alpha_n)_{n \in [m, +\infty[}$ a complex number sequence such that $\alpha_0 = 0$, and such that for $0 < \theta \ll 1$:

$$ \partial_s J_{\mathbf{N},\mathbf{N}'}(\theta, s) = \sum_{\mathcal{P} \subseteq [1,P]} \sum_{\substack{\mathbf{k}=(k_p)_{p \in \mathcal{P}^c} \in \mathbb{N}_0^{\mathcal{P}^c} \\ |\mathbf{k}|=|\mathbf{N}'|+|\mathbf{N}_{||\mathcal{P}|+||\mathcal{P}|}|}} \left( \prod_{p \in \mathcal{P}} N_p! \right) (-1)^{|\mathbf{k}|+N'_j} \frac{\mu'_{j}}{|\boldsymbol{\mu}'|+|\boldsymbol{\mu}|_{||\mathcal{P}|}} N'_j! \prod_{p \in \mathcal{P}^c} \frac{\zeta(-N_p - k_p, d_p)}{k_p!} \tag{5.13} $$

$$
\begin{aligned}
& \cdot \left( Q_{\mathbf{N},\mathbf{N}'}^0(j, \mathcal{P}, \mathbf{k}) \left( \sum_{p \in \mathcal{P}} \mu_p(\gamma - h_{N_p}) + \mu'_j(\gamma - h_{N'_j}) \right) + Q_{\mathbf{N},\mathbf{N}'}^1(j, \mathcal{P}, \mathbf{k}) \right) \\
& + \sum_{\mathcal{P} \subseteq [1,P]} \sum_{j=1}^Q \left( \prod_{p \in \mathcal{P}} N_p! \right) (-1)^{|\mathbf{N}'|+|\mathbf{N}_{||\mathcal{P}|+||\mathcal{P}|}|+N'_j} \frac{\mu'_j N'_j!}{|\boldsymbol{\mu}'|+|\boldsymbol{\mu}|_{||\mathcal{P}|}} \sum_{\substack{\mathbf{k}=(k_p)_{p \in \mathcal{P}^c} \in \mathbb{N}_0^{\mathcal{P}^c} \\ |\mathbf{k}|=|\mathbf{N}'|+|\mathbf{N}_{||\mathcal{P}|+||\mathcal{P}|}|}} Q_{\mathbf{N},\mathbf{N}'}^0(j, \mathcal{P}, \mathbf{k}) \\
& \cdot \left( \sum_{i \in \mathcal{P}^c} \mu_i \frac{\zeta'(-N_i - k_i, d_i)}{k_i!} \prod_{p \in \mathcal{P}^c \setminus \{i\}} \frac{\zeta(-N_p - k_p, d_p)}{k_p!} \right) \\
& + J_{\mathbf{N},\mathbf{N}'}^*(\theta)_{\mu,\mu'}
\end{aligned}
$$
*Proof.* Let's recall that

$$Q_{\mathbf{N},\mathbf{N}'}^{0}(j, \mathcal{P}, \mathbf{k}) := h_{\mathbf{N},\mathbf{N}',\mathcal{P},j,\mathbf{k}}(0)$$

$$Q_{\mu, \mu'}^{1}(j, \mathcal{P}, \mathbf{k}) := \partial_s (h_{\mathbf{N},\mathbf{N}',\mathcal{P},j,\mathbf{k}}(s))|_{s=0},$$

for $j \in [1, Q]$, $\mathcal{P} \subsetneq [1, P]$, $\mathbf{k} = (k_p)_{p \in \mathcal{P}} \in \mathbb{N}_0^{\mathcal{P}^c}$.

Let's note $T_{\mathbf{N},\mathbf{N}',\mathcal{P},j,\mathbf{k}}(\theta, s)$ the general term in the series in formula 2.5: for $\mu, \mu'$

$$T_{\mathbf{N},\mathbf{N}',\mathcal{P},j,\mathbf{k}}(\theta, s) = \frac{\left( \prod_{p \in \mathcal{P}} \Gamma(1 + N_p - \mu_p s) \right) \sum_{\mathbf{k}=(k_p)_{p \in \mathcal{P}} \in \mathbb{N}_0^{\mathcal{P}^c}} (-1)^{|\mathbf{k}|} h_{\mathbf{N},\mathbf{N}',\mathcal{P},j,\mathbf{k}}(s) \theta^{(|\mu'|+|\mu|_P)s - |\mathbf{N}'|-|\mathbf{N}|_P - |\mathcal{P}|-|\mathbf{k}|} {\Gamma(-N'_j + \mu'_j s) ((|\mu'| + |\mu|_P)s - |\mathbf{N}'| - |\mathbf{N}|_P - |\mathcal{P}| + |\mathbf{k}|) \prod_{p \in \mathcal{P}^c} \frac{\zeta(-N_p + \mu_p s - k_p, d_p)}{k_p!}},$$

for $j \in [1, Q]$, $\mathcal{P} \subsetneq [1, P]$, and $\mathbf{k} = (k_p)_{p \in \mathcal{P}} \in \mathbb{N}_0^{\mathcal{P}^c}$. We're now going to evaluate formula 2.29 at $s=0$ in order to get the desired formula for $\partial_s(J_{\mathbf{N},\mathbf{N}'}(\theta, s))_{s=0}$. We get

$$\partial_s(J_{\mathbf{N},\mathbf{N}'}(\theta, s)) = \sum_{\mathcal{P} \subsetneq [1,P]} \sum_{j=1}^{Q} \sum_{\mathbf{k}=(k_p)_{p \in \mathcal{P}} \in \mathbb{N}_0^{\mathcal{P}^c}} \partial_s T_{\mathbf{N},\mathbf{N}',\mathcal{P},j,\mathbf{k}}(\theta, s))_{s=0} \quad (5.14) \\ + \partial_s \left( \prod_{p=1}^{P} \Gamma(1 + N_p - \mu_p s) \sum_{j=1}^{Q} h_{\mathbf{N},\mathbf{N}',[1,P],j}(s) \frac{\theta^{(|\mu'|+|\mu|)s - |\mathbf{N}'|-|\mathbf{N}|-P}}{\Gamma(-N'_j + \mu'_j s) ((|\mu'| + |\mu|)s - |\mathbf{N}'| - |\mathbf{N}|-P)} \right)_{s=0}.$$

Firstly, lets study the second term of this sum. We see that, at $s=0$, the function $\frac{1}{\Gamma(-N'_j + \mu'_j s)((|\mu'| + |\mu|)s - |\mathbf{N}'| - |\mathbf{N}|-P)}$ vanish. Therefore, we get that

$$\partial_s \left( \prod_{p=1}^{P} \Gamma(1 + N_p - \mu_p s) \sum_{j=1}^{Q} h_{\mathbf{N},\mathbf{N}',[1,P],j}(s) \frac{\theta^{(|\mu'|+|\mu|)s - |\mathbf{N}'|-|\mathbf{N}|-P}}{\Gamma(-N'_j + \mu'_j s)((|\mu'| + |\mu|)s - |\mathbf{N}'| - |\mathbf{N}|-P)} \right)_{s=0} = \prod_{p=1}^{P} N_p! \sum_{j=1}^{Q} h_{\mathbf{N},\mathbf{N}',[1,P],j}(0) \theta^{-|\mathbf{N}'|-|\mathbf{N}|-P} \partial_s \left( \frac{1}{\Gamma(-N'_j + \mu'_j s)((|\mu'| + |\mu|)s - |\mathbf{N}'| - |\mathbf{N}|-P)} \right)_{s=0}.$$

By Lemma 3.1.1, we have

$$\partial_s \left( \frac{1}{\Gamma(-N'_j + \mu'_j s)((|\mu'| + |\mu|)s - |\mathbf{N}'| - |\mathbf{N}|-P)} \right)_{s=0} = (-1)^{N'_j+1} N'_j! \frac{\mu'_j}{|\mathbf{N}'| + |\mathbf{N}| + P}.$$

We then find that

$$\partial_s \left( \prod_{p=1}^{P} \Gamma(1 + N_p - \mu_p s) \sum_{j=1}^{Q} h_{\mathbf{N},\mathbf{N}',[1,P],j}(s) \frac{\theta^{(|\mu'|+|\mu|)s - |\mathbf{N}'|-|\mathbf{N}|-P}}{\Gamma(-N'_j + \mu'_j s)((|\mu'| + |\mu|)s - |\mathbf{N}'| - |\mathbf{N}|-P)} \right)_{s=0} = \left( \prod_{p=1}^{P} N_p! \sum_{j=1}^{Q} h_{\mathbf{N},\mathbf{N}',[1,P],j}(0) (-1)^{N'_j+1} N'_j! \frac{\mu'_j}{|\mathbf{N}'| + |\mathbf{N}| + P} \right) \theta^{-|\mathbf{N}'|-|\mathbf{N}|-P}.$$

We shall include later on that last term in the definition $J_{\mathbf{N},\mathbf{N}'}^*(\theta)$, because $-|\mathbf{N}'| - |\mathbf{N}|-P \neq 0$. For $\mu, \mu'$
Let's now study the derivative of the general term $T_{\mathbf{N},\mathbf{N}',\mathcal{P},j,\mathbf{k}}(\theta, s))$ with respect to $s$ at $s = 0$.

• If $|\mathbf{k}| \neq |\mathbf{N}'| + |\mathbf{N}|_{\mathcal{P}} + |\mathcal{P}|$, then the function $\frac{1}{\Gamma(-N'_j + \mu'_j s)((|\boldsymbol{\mu}'| + |\boldsymbol{\mu}|_{\mathcal{P}})s - |\mathbf{N}'| - |\mathbf{N}|_{\mathcal{P}} - |\mathcal{P}| + |\mathbf{k}|)}$ vanish at $s = 0$. Therefore we find that

$$ \partial_s (T_{\mathbf{N},\mathbf{N}',\mathcal{P},j,\mathbf{k}}(\theta, s))_{s=0} = \left( \prod_{p \in \mathcal{P}} N_p! \right) (-1)^{|\mathbf{k}|} h_{\mathbf{N},\mathbf{N}',\mathcal{P},j,\mathbf{k}}(0) \theta^{-|\mathbf{N}'|-|\mathbf{N}|_{\mathcal{P}}-|\mathcal{P}|+|\mathbf{k}|} \partial_s \left( \frac{1}{\Gamma(-N'_j + \mu'_j s)((|\boldsymbol{\mu}'| + |\boldsymbol{\mu}|_{\mathcal{P}})s - |\mathbf{N}'| - |\mathbf{N}|_{\mathcal{P}} - |\mathcal{P}| + |\mathbf{k}|)} \right)_{s=0} \prod_{p \in \mathcal{P}^c} \frac{\zeta(-N_p + \mu_p s - k_p, d_p)}{k_p!} $$

Via Lemma 3.1.1, we get

$$ \partial_s (T_{\mathbf{N},\mathbf{N}',\mathcal{P},j,\mathbf{k}}(\theta, s))_{s=0} = \left( \prod_{p \in \mathcal{P}} N_p! \right) (-1)^{|\mathbf{k}|} h_{\mathbf{N},\mathbf{N}',\mathcal{P},j,\mathbf{k}}(0) \theta^{-|\mathbf{N}'|-|\mathbf{N}|_{\mathcal{P}}-|\mathcal{P}|+|\mathbf{k}|} \frac{(-1)^{N'_j} N'_j!}{-|\mathbf{N}'|-|\mathbf{N}|_{\mathcal{P}}-|\mathcal{P}|+|\mathbf{k}|} \prod_{p \in \mathcal{P}^c} \frac{\zeta(-N_p + \mu_p s - k_p, d_p)}{k_p!} $$

We see that the power of $\theta$ in the last term is non-zero. Therefore we shall later on include the terms $\partial_s (T_{\mathbf{N},\mathbf{N}',\mathcal{P},j,\mathbf{k}}(\theta, s))_{s=0}$ such that $|\mathbf{k}| \neq |\mathbf{N}'| + |\mathbf{N}|_{\mathcal{P}} + |\mathcal{P}|$ into the definition of $J^*_{\mathbf{N},\mathbf{N}'}(\theta)_{\mu,\mu'}$.

• If $|\mathbf{k}| = |\mathbf{N}'| + |\mathbf{N}|_{\mathcal{P}} + |\mathcal{P}|$, then Lemma 3.1.1, we have

$$ \left( \frac{1}{\Gamma(-N'_j + \mu'_j s)((|\boldsymbol{\mu}'| + |\boldsymbol{\mu}|_{\mathcal{P}})s - |\mathbf{N}'| - |\mathbf{N}|_{\mathcal{P}} - |\mathcal{P}| + |\mathbf{k}|)} \right)_{s=0} = (-1)^{N'_j} \frac{\mu'_j}{|\boldsymbol{\mu}'| + |\boldsymbol{\mu}|_{\mathcal{P}}} N'_j! \\ \partial_s \left( \frac{1}{\Gamma(-N'_j + \mu'_j s)((|\boldsymbol{\mu}'| + |\boldsymbol{\mu}|_{\mathcal{P}})s - |\mathbf{N}'| - |\mathbf{N}|_{\mathcal{P}} - |\mathcal{P}| + |\mathbf{k}|)} \right)_{s=0} = (-1)^{N'_j} \frac{\mu_j'^2}{|\boldsymbol{\mu}'| + |\boldsymbol{\mu}|_{\mathcal{P}}} N_j'! (\gamma - h_{N'_j}). $$

Taking the derivative with respect to $s$ of the general term $T_{\mathbf{N},\mathbf{N}',\mathcal{P},j,\mathbf{k}}(\theta, s)$, we get

$$ \partial_s (T_{\mathbf{N},\mathbf{N}',\mathcal{P},j,\mathbf{k}}(\theta, s))_{s=0} = - \left( \sum_{i \in \mathcal{P}} \mu_i \Gamma'(1 + N_i) \prod_{p \in \mathcal{P} \setminus \{i\}} N_p! \right) (-1)^{|\mathbf{k}|} Q^0_{\mathbf{N},\mathbf{N}'}(j, \mathcal{P}, \mathbf{k}) (-1)^{N'_j} \frac{\mu'_j}{|\boldsymbol{\mu}'| + |\boldsymbol{\mu}|_{\mathcal{P}}} N'_j! \prod_{p \in \mathcal{P}^c} \frac{\zeta(-N_p - k_p, d_p)}{k_p!} \\ + \left( \prod_{p \in \mathcal{P}} N_p! \right) (-1)^{|\mathbf{k}|} Q^1_{\mathbf{N},\mathbf{N}'}(j, \mathcal{P}, \mathbf{k}) (-1)^{N'_j} \frac{\mu'_j}{|\boldsymbol{\mu}'| + |\boldsymbol{\mu}|_{\mathcal{P}}} N'_j! \prod_{p \in \mathcal{P}^c} \frac{\zeta(-N_p - k_p, d_p)}{k_p!} \\ + \left( \prod_{p \in \mathcal{P}} N_p! \right) (-1)^{|\mathbf{k}|} Q^0_{\mathbf{N},\mathbf{N}'}(j, \mathcal{P}, \mathbf{k}) (|\boldsymbol{\mu}'| + |\boldsymbol{\mu}|_{\mathcal{P}}) \ln(\theta) (-1)^{N'_j} \frac{\mu'_j}{|\boldsymbol{\mu}'| + |\boldsymbol{\mu}|_{\mathcal{P}}} N'_j! \prod_{p \in \mathcal{P}^c} \frac{\zeta(-N_p - k_p, d_p)}{k_p!} \\ + \left( \prod_{p \in \mathcal{P}} N_p! \right) (-1)^{|\mathbf{k}|} Q^0_{\mathbf{N},\mathbf{N}'}(j, \mathcal{P}, \mathbf{k}) (-1)^{N'_j} \frac{\mu_j'^2}{|\boldsymbol{\mu}'| + |\boldsymbol{\mu}|_{\mathcal{P}}} N_j'! (\gamma - h_{N'_j}) \prod_{p \in \mathcal{P}^c} \frac{\zeta(-N_p - k_p, d_p)}{k_p!} \\ + \left( \prod_{p \in \mathcal{P}} N_p! \right) (-1)^{|\mathbf{k}|} Q^0_{\mathbf{N},\mathbf{N}'}(j, \mathcal{P}, \mathbf{k}) (-1)^{N'_j} \frac{\mu'_j}{|\boldsymbol{\mu}'| + |\boldsymbol{\mu}|_{\mathcal{P}}} N'_j! \left( \sum_{i \in \mathcal{P}^c} \mu_i \frac{\zeta'(-N_i - k_i, d_i)}{k_i!} \prod_{p \in \mathcal{P}^c \setminus \{i\}} \frac{\zeta(-N_p - k_p, d_p)}{k_p!} \right), $$
We know that $\Gamma'(1 + N_p) = N'_p(-\gamma + h_{N'_p})$. Therefore, we can simplify the last equality:

$$
\begin{align*}
\partial_s (T_{\mathbf{N},\mathbf{N}',\mathcal{P},j,\mathbf{k}}(\theta, s))_{s=0} = & \left( \prod_{p \in \mathcal{P}} N_p! \right) (-1)^{|\mathbf{k}|+N'_j} \frac{\mu'_j N'_j!}{|\boldsymbol{\mu}'| + |\boldsymbol{\mu}|_{\mathcal{P}}} \prod_{p \in \mathcal{P}^c} \frac{\zeta(-N_p - k_p, d_p)}{k_p!} \\
& \cdot \left( Q_{\mathbf{N},\mathbf{N}'}^0 (j, \mathcal{P}, \mathbf{k}) \left( \mu'_j(\gamma - h_{N'_j}) - \sum_{p \in \mathcal{P}} \frac{\mu_p \Gamma'(1+N_p)}{N_p!} \right) + Q_{\mathbf{N},\mathbf{N}'}^1 (j, \mathcal{P}, \mathbf{k}) \right) \\
& + \left( \prod_{p \in \mathcal{P}} N_p! \right) (-1)^{|\mathbf{k}|} Q_{\mathbf{N},\mathbf{N}'}^0 (j, \mathcal{P}, \mathbf{k}) (-1)^{N'_j} \frac{\mu'_j N'_j!}{|\boldsymbol{\mu}'| + |\boldsymbol{\mu}|_{\mathcal{P}}} \left( \sum_{i \in \mathcal{P}^c} \mu_i \frac{\zeta'(-N_i - k_i, d_i)}{k_i!} \prod_{p \in \mathcal{P}^c \setminus \{i\}} \frac{\zeta(-N_p - k_p, d_p)}{k_p!} \right), \\
& + \left( \left( \prod_{p \in \mathcal{P}} N_p! \right) (-1)^{|\mathbf{k}|+N'_j} Q_{\mathbf{N},\mathbf{N}'}^0 (j, \mathcal{P}, \mathbf{k}) \mu'_j N'_j! \prod_{p \in \mathcal{P}^c} \frac{\zeta(-N_p - k_p, d_p)}{k_p!} \right) \ln(\theta).
\end{align*}
$$

We shall later on include the terms of the form $\ln(\theta)$ into the definition of $J^*_{\mathbf{N},\mathbf{N}'}(\theta)$.

We now define $J^*_{\mathbf{N},\mathbf{N}'}(\theta)$ such that
$$J^*_{\mu,\mu'} = J^*_{\mu',\mu}(\theta)$$

$$
\begin{align*}
J^*_{\mu, \mu'}(\theta) = & \left( \sum_{\mathcal{P} \subseteq [1, P]} \left( \prod_{p \in \mathcal{P}} N_p! \right) \sum_{j=1}^{Q} \mu'_j N'_j! \sum_{\substack{k=(k_p)_{p \in \mathcal{P}^c} \in \mathbb{N}_0^{\mathcal{P}^c} \\ |\mathbf{k}|=|\mathbf{N}'|+|\mathbf{N}|_{\mathcal{P}}+|\mathcal{P}|}} (-1)^{|\mathbf{k}|+N'_j} Q_{\mathbf{N},\mathbf{N}'}^0 (j, \mathcal{P}, \mathbf{k}) \prod_{p \in \mathcal{P}^c} \frac{\zeta(-N_p - k_p, d_p)}{k_p!} \right) \ln(\theta) \\
& + \sum_{\mathcal{P} \subseteq [1, P]} \sum_{j=1}^{Q} \sum_{\substack{k=(k_p)_{p \in \mathcal{P}^c} \in \mathbb{N}_0^{\mathcal{P}^c} \\ |\mathbf{k}| \neq |\mathbf{N}'|+|\mathbf{N}|_{\mathcal{P}}+|\mathcal{P}|}} \left( \left( \prod_{p \in \mathcal{P}} N_p! \right) (-1)^{|\mathbf{k}|} Q_{\mathbf{N},\mathbf{N}'}^0 (j, \mathcal{P}, \mathbf{k}) (-1)^{N'_j} N'_j! \frac{\mu'_j}{-|\mathbf{N}'| - |\mathbf{N}|_{\mathcal{P}} - |\mathcal{P}| + |\mathbf{k}|} \right) \\
& \cdot \prod_{p \in \mathcal{P}^c} \frac{\zeta(-N_p + \mu_p s - k_p, d_p)}{k_p!} \theta^{-|\mathbf{N}'|-|\mathbf{N}|_{\mathcal{P}}-|\mathcal{P}|+|\mathbf{k}|} \\
& + \left( \prod_{p=1}^{P} N_p! \sum_{j=1}^{Q} Q_{\mathbf{N},\mathbf{N}'}^0 (j, [1, P]) (-1)^{N'_j+1} N'_j! \frac{\mu'_j}{|\mathbf{N}'| + |\mathbf{N}| + P} \right) \theta^{-|\mathbf{N}'|-|\mathbf{N}|-P}.
\end{align*}
$$

By cutting off the series in formula 5.14 whether if $|\mathbf{k}|$ is equal or not to $|\mathbf{N}'| + |\mathbf{N}|_{\mathcal{P}} + |\mathcal{P}|$, and by definition of the term $J^*_{\mathbf{N},\mathbf{N}'}(\theta)$ that we just defined, we get the required formula 5.13. $\square$

We see that the series $J^*_{\mathbf{N},\mathbf{N}'}(\theta)$ and $K^*_{\mathbf{N},\mathbf{N}'}(\theta)$ have a zero constant term, and that $\partial_s Z_{\mathbf{N},\mathbf{N}'}(s) = \partial_s J_{\mathbf{N},\mathbf{N}'}(\theta, s) + \partial_s K_{\mathbf{N},\mathbf{N}'}(\theta, s)$. As $\partial_s (Z_{\mathbf{N},\mathbf{N}'}(s))_{s=0}$ does not depend on $\theta$, by Lemma 3.1.3 we have

$$
\partial_s (Z_{\mathbf{N},\mathbf{N}'}(s))|_{s=0} = \partial_s J_{\mathbf{N},\mathbf{N}'}(\theta, s)|_{s=0} + \partial_s (K_{\mathbf{N},\mathbf{N}'}(\theta, s))|_{s=0} - K^*_{\mathbf{N},\mathbf{N}'}(\theta) - J^*_{\mathbf{N},\mathbf{N}'}(\theta)
$$

for every $0 < \theta \ll 1$. This leads us to Theorem D.

At the same time, we also obtain Theorem C, which corresponds to a qualitative reformulation of
Theorem D.

**Remark 5.3.3.** From the equality $K^*_{\mathbf{N},\mathbf{N}'}(\theta) + J^*_{\mathbf{N},\mathbf{N}'}(\theta) = 0$ for all $0 < \theta \ll 1$, and from Lemma 3.1.3, we get that each coefficient in the Laurent series in $K^*_{\mathbf{N},\mathbf{N}'}(\theta) + J^*_{\mathbf{N},\mathbf{N}'}(\theta)$ vanishes, and that the coefficient
in front of $\ln(\theta)$ also vanishes. We can then get new relations between the special values $\zeta(-k, d_p)$, with $k \in \mathbb{N}_0$ and $1 \le p \le P$.

## 5.4 On the corollaries of Theorem D

To prove the corollaries of Theorem D, we will use the formula given by the theorem, and substitute the direction or the values $-(\mathbf{N}, \mathbf{N}')$ that one wishes to study.

### 5.4.1 Proof of Theorem C

Let $(\mathbf{N}, \mathbf{N}') \in \mathbb{N}_0^P \times \mathbb{N}_0^Q$, and $(\mu, \mu') \in \mathbb{R}^{+P} \times \mathbb{R}_*^{+Q}$. The goal here is to revisit the formula obtained for $Z'(-(\mathbf{N}, \mathbf{N}'))$ and to study its terms qualitatively. We denote $\mathbb{K}$ as the field generated above $\mathbb{Q}$ by the complex coefficients $(d_p)_{p \in [1, P]}$ and $(c_{q,p})_{p \in [1, P], q \in [1, Q]}$, and the directions $(\mu_p)_{p \in [1, P]}$, $(\mu'_q)_{q \in [1, Q]}$. By Theorem D, we have

$$
\begin{aligned}
Z'(-(\mathbf{N}, \mathbf{N}')) = & \sum_{\mathcal{P} \subseteq [1, P]} \sum_{j=1}^{Q} (-1)^{|\mathbf{N}|+|\mathbf{N}'|+|\mathcal{P}|+N'_j} \left( \prod_{p \in \mathcal{P}} N_p! \right) \frac{\mu'_j N'_j!}{|\boldsymbol{\mu}'| + |\boldsymbol{\mu}|_{\mathcal{P}}} \sum_{\substack{\mathbf{k}=(k_p)_{p \in \mathcal{P}} \in \mathbb{N}_0^{P_c} \\ |\mathbf{k}|=|\mathbf{N}'|+|\mathbf{N}_{||P}|-|\mathcal{P}|}} \prod_{p \in \mathcal{P}} \frac{\zeta(-N_p - k_p, d_p)}{k_p!} \\
& \cdot \left( Q_{\mathbf{N}, \mathbf{N}'}^0(j, \mathcal{P}, \mathbf{k}) \left( \sum_{p \in \mathcal{P}} \mu_p(\gamma - h_{N_p}) + \mu'_j(\gamma - h_{N'_j}) \right) + Q_{\mathbf{N}, \mathbf{N}'}^1(j, \mathcal{P}, \mathbf{k}) \right) \\
& + \sum_{\mathcal{P} \subseteq [1, P]} \sum_{j=1}^{Q} \left( \prod_{p \in \mathcal{P}} N_p! \right) (-1)^{|\mathbf{N}'|+|\mathbf{N}_{||P}|-|\mathcal{P}|+N'_j} \frac{\mu'_j N'_j!}{|\boldsymbol{\mu}'| + |\boldsymbol{\mu}|_{\mathcal{P}}} \sum_{\substack{\mathbf{k}=(k_p)_{p \in \mathcal{P}} \in \mathbb{N}_0^{P_c} \\ |\mathbf{k}|=|\mathbf{N}'|+|\mathbf{N}_{||P}|-|\mathcal{P}|}} Q_{\mathbf{N}, \mathbf{N}'}^0(j, \mathcal{P}, \mathbf{k}) \\
& \cdot \left( \sum_{i \in \mathcal{P}^c} \mu_i \frac{\zeta'(-N_i - k_i, d_i)}{k_i!} \prod_{p \in \mathcal{P}^c \setminus \{i\}} \frac{\zeta(-N_p - k_p, d_p)}{k_p!} \right) \\
& + \sum_{j=1}^{Q} \mu'_j \sum_{\substack{\mathbf{u}=(\mathbf{u}_q)_{q \in [1, Q] \setminus \{j\}} \in (\mathbb{N}_0^P)^{[1, Q] \setminus \{j\}} \\ \forall q \in [1, Q] \setminus \{j\}, |\mathbf{u}_q|=N'_q}} \left( \prod_{q=1}^{Q} \binom{N'_q}{q \ne j} \prod_{p=1}^{P} c_{q,p}^{u_{q,p}} \right) \partial_s \varphi_{\mathbf{R}(\mathbf{N}, j, \mathbf{u})}^j(s)|_{s=-N'_j} \\
& - \sum_{j=1}^{Q} (\gamma - h_{N'_j}) \mu'_j N'_j! \sum_{\substack{\mathbf{u}=(\mathbf{u}_q)_{q \in [1, Q] \setminus \{j\}} \in (\mathbb{N}_0^P)^{[1, Q] \setminus \{j\}} \\ \forall q \in [1, Q] \setminus \{j\}, |\mathbf{u}_q|=N'_q}} \left( \left( \prod_{q=1}^{Q} \binom{N'_q}{q \ne j} \prod_{p=1}^{P} c_{q,p}^{u_{q,p}} \right) \partial_s \varphi_{\mathbf{R}(\mathbf{N}, j, \mathbf{u})}^j(s) \right) |_{s=-N'_j} \\
& \cdot \left( \sum_{\mathcal{P} \subseteq [1, P]} (-1)^{|\mathbf{R}(\mathbf{N}, j, \mathbf{u})|_{||P}+|\mathcal{P}|} \prod_{p \in \mathcal{P}} c_{j,p}^{-R(\mathbf{N}, j, \mathbf{u})_p-1} R(\mathbf{N}, j, \mathbf{u})_p! \right) \sum_{\substack{\mathbf{k}=(k_p)_{p \in \mathcal{P}} \in \mathbb{N}_0^{P_c} \\ |\mathbf{k}|=N'_j+|\mathbf{R}(\mathbf{N}, j, \mathbf{u})|_{||P}+|\mathcal{P}|}} \prod_{p \in \mathcal{P}} c_{j,p}^{k_p} \frac{\zeta(-\mathbf{R}(\mathbf{N}, j, \mathbf{u})_p - k_p, d_p)}{k_p!}
\end{aligned}
$$

where we have set

$$ \mathbf{R}(\mathbf{N}, j, \mathbf{u}) := (N_p + \mathbf{u}(p))_{p \in [1, P]}, $$