Si ce terme est strictement négatif, alors le coefficient binomial $\left( \frac{-N-1}{\sum_{q \neq j} (N - w_{2,q})} \right)$ est nul. Ainsi, on peut imposer la condition $w_{2,j} \ge 2N+1$ :

$$Q_N^0(\{1\}, k_2) = (-1)^{N(Q-1)} N!^{Q-1} \sum_{\substack{j=1 \\ |\mathbf{w}_2|=(Q+1)N+1 \\ w_{2,j} \ge 2N+1}}^{Q} \sum_{\substack{\mathbf{w}_2 \in \mathbb{N}^Q \\ |\mathbf{w}_1|=(Q+1)N+1 \\ w_{1,j} \ge 2N+1}} \frac{(-N-1)...(N+1-w_{2,j})}{\prod_{q \neq j} w_{2,q}!} c_{j,2}^{w_{2,j}} \prod_{\substack{q=1 \\ q \neq j}}^{Q} c_{q,1}^{N-w_{2,q}} c_{q,2}^{w_{2,q}}.$$

• Si $\mathcal{P} = \{2\}$ et $k_1 = (Q+1)N+1$, alors on a

$$Q_N^0(\{2\},) = (-1)^{N(Q-1)} N!^{Q-1} \sum_{\substack{j=1 \\ |\mathbf{w}_1|=(Q+1)N+1 \\ w_{1,j} \ge 2N+1}}^{Q} \sum_{\substack{\mathbf{w}_1 \in \mathbb{N}^Q \\ |\mathbf{w}_2|=(Q+1)N+1 \\ w_{2,j} \ge 2N+1}} \frac{(-N-1)...(N+1-w_{1,j})}{\prod_{q \neq j} w_{1,q}!} c_{j,1}^{w_{1,j}} \prod_{\substack{q=1 \\ q \neq j}}^{Q} c_{q,2}^{N-w_{1,q}} c_{q,1}^{w_{1,q}}.$$

□

Via le lemme précédent et en utilisant le fait que $\zeta(-n) = (-1)^{n+1} \frac{B_{n+1}}{n+1}$, on peut simplifier les expressions pour qu'elles soient plus pratiques à calculer explicitement :

$$Z_{\Delta}(-N) = \frac{(-1)^{QN}}{Q} N!^Q \sum_{k=0}^{QN} \alpha_k(N) \frac{B_{N+k+1}}{(N+k+1)k!} \frac{B_{(Q+1)N-k+1}}{((Q+1)N-k+1)(QN-k)!} \quad (6.2)$$

avec

$$\alpha_k(N) = \sum_{j=1}^{Q} \sum_{\substack{(w_q)_{q \in \{1, Q\}} \in [0, N]^Q \\ w_1 + \dots + w_Q = k}} \binom{k}{(w_1, \dots, w_Q)} \binom{QN-k}{(N-w_1, \dots, N-w_Q)} \prod_{q=1}^{Q} c_{q,1}^{w_q} c_{q,2}^{N-w_q},$$

$$\beta(N) = \sum_{\substack{j=1 \\ w_1 + \dots + w_Q = (Q+1)N+1 \\ w_j \ge 2N+1}}^{Q} \sum_{\substack{\mathbf{w}_1 \in \{1, Q\} \\ w_1 + \dots + w_Q = (Q+1)N+1 \\ w_j \ge 2N+1}} \frac{(-N-1)...(N-1-w_j)}{\prod_{q \neq j} w_q!} \left( c_{j,2}^{w_j} \prod_{\substack{q=1 \\ q \neq j}}^{Q} c_{q,1}^{N-w_q} c_{q,2}^{w_q} + c_{j,1}^{w_j} \prod_{\substack{q=1 \\ q \neq j}}^{Q} c_{q,2}^{N-w_q} c_{q,1}^{w_q} \right).$$

### Lemme 6.3.2.

$$Z'_{\Delta}(0) = Q\zeta(0)^2\gamma - \zeta(-1) \left( \sum_{j=1}^{Q} (c_{j,1} + c_{j,2}) \right) \gamma \\ - \frac{1}{Q+1}\zeta(-1) \sum_{\substack{j=1 \\ f \neq j}}^{Q} \sum_{\substack{f=1 \\ f \neq j}}^{Q} \left( \frac{1}{c_{j,2}}(c_{f,1} - c_{j,1}) \ln\left(1 + \frac{c_{f,2}}{c_{j,2}}\right) + \frac{1}{c_{j,1}}(c_{f,2} - c_{j,2}) \ln\left(1 + \frac{c_{f,1}}{c_{j,1}}\right) \right) \\ + \frac{\zeta'(0)\zeta(0)}{Q} - \frac{\zeta'(-1)}{Q+1} \sum_{j=1}^{Q} (c_{j,1} + c_{j,2}) - \gamma\zeta(0)^2 \sum_{j=1}^{Q} (c_{j,1}c_{j,2})^{-1} + \gamma\zeta(-1) \sum_{j=1}^{Q} (c_{j,1}^{-1}c_{j,2} + c_{j,2}^{-1}c_{j,1}) \\ + \sum_{j=1}^{Q} \partial_s \zeta^B(s, d'_j|c_{j,1}, c_{j,2})|_{s=0}.$$
Démonstration. Rappelons que $Z'_{\Delta}(0) = Z'(-(0,0))$ Comme on a posé $P = 2$ et que l'on a fixé les directions $(\boldsymbol{\mu}, \boldsymbol{\mu}') = (\mathbf{1}, \mathbf{1})$, on a par le Corollaire D2 :

$$
\begin{align*}
Z'(-(\mathbf{0}, \mathbf{0})) ={}& \sum_{\mathcal{P} \subseteq [1,2]} (-1)^{|\mathcal{P}|} \frac{1}{Q + |\mathcal{P}|} \sum_{\mathbf{k}=(k_p)_{p \in \mathcal{P}^c} \in \mathbb{N}^{\mathcal{P}^c} \atop |\mathbf{k}| = |\mathcal{P}|} \prod_{p \in \mathcal{P}^c} \frac{\zeta(-k_p)}{k_p!} \sum_{j=1}^{Q} \left( \left( \prod_{p \in \mathcal{P}^c} c_{j,p}^{k_p} \right) \gamma(1 + |\mathcal{P}|) + Q^1(j, \mathcal{P}, \mathbf{k}) \right) \\
& + \sum_{\mathcal{P} \subseteq [1,2]} \sum_{j=1}^{Q} (-1)^{|\mathcal{P}|} \frac{1}{Q + |\mathcal{P}|} \sum_{\mathbf{k}=(k_p)_{p \in \mathcal{P}^c} \in \mathbb{N}^{\mathcal{P}^c} \atop |\mathbf{k}| = |\mathcal{P}|} \prod_{p \in \mathcal{P}^c} c_{j,p}^{k_p} \left( \sum_{i \in \mathcal{P}^c} \frac{\zeta'(-k_i)}{k_i!} \prod_{p \in \mathcal{P}^c \setminus \{i\}} \frac{\zeta(-k_p)}{k_p!} \right) \\
& + \sum_{j=1}^{Q} \partial_s \zeta^B(s, d'_j | c_{j,1}, c_{j,2})|_{s=0} - \sum_{j=1}^{Q} \gamma \sum_{\mathcal{P} \subseteq [1,2]} (-1)^{|\mathcal{P}|} \prod_{p \in \mathcal{P}} c_{j,p}^{-1} \sum_{\mathbf{k}=(k_p)_{p \in \mathcal{P}^c} \in \mathbb{N}^{\mathcal{P}^c} \atop |\mathbf{k}| = |\mathcal{P}|} \prod_{p \in \mathcal{P}^c} c_{j,p}^{k_p} \frac{\zeta(-k_p)}{k_p!},
\end{align*}
$$

avec

$$
Q^1(j, \mathcal{P}, \mathbf{k}) = \sum_{\substack{f=1 \\ f \neq j}}^{Q} \sum_{\mathbf{w}'=(w'_p)_{p \in \mathcal{P}^c} \in \prod_{p \in \mathcal{P}^c} [0, k_p]} F_{\mathcal{P}, j, f, \mathbf{0}, (w'_p \mathbf{e}_f + (k_p - w'_p) \mathbf{e}_j)_{p \in \mathcal{P}^c}} (\mathbf{0}, \mathbf{0}) \left( \prod_{p \in \mathcal{P}^c} \binom{k_p}{w'_p} c_{f,p}^{w'_p} c_{j,p}^{k_p - w'_p} \right) \\
+ (Q-1) \gamma \prod_{p \in \mathcal{P}^c} c_{j,p}^{k_p}.
$$

en rappelant la notation $\mathbf{e}_n = (0, \dots, 1, \dots, 0) \in \mathbb{R}^Q$ avec un 1 sur la $n$-ième composante et des 0 ailleurs.
En développant la somme présente dans l'expression de $Z'(-(\mathbf{0}, \mathbf{0}))$, on obtient
$$
Z'(-(\mathbf{0}, \mathbf{0})) =
$$
$$
\begin{aligned}
& Z'(-(\mathbf{0}, \mathbf{0})) = \\
& \quad \zeta(0)^2 \gamma - \frac{2\zeta(-1)}{Q+1} \gamma \left( \sum_{j=1}^{Q} (c_{j,1} + c_{j,2}) \right) + \frac{1}{Q} \zeta(0)^2 \sum_{j=1}^{Q} Q^1(j, \emptyset, (0,0)) \\
& \quad - \frac{1}{Q+1} \zeta(-1) \sum_{j=1}^{Q} (Q^1(j, \{1\}, 1) + Q^1(j, \{2\}, 1)) + \frac{\zeta'(0)\zeta(0)}{Q} - \frac{\zeta'(-1)}{Q+1} \sum_{j=1}^{Q} (c_{j,1} + c_{j,2}) \\
& \quad + \sum_{j=1}^{Q} \partial_s \zeta^B(s, d'_j | c_{j,1}, c_{j,2})|_{s=0} - \gamma \zeta(0)^2 \sum_{j=1}^{Q} (c_{j,1} c_{j,2})^{-1} + \gamma \zeta(-1) \sum_{j=1}^{Q} (c_{j,1}^{-1} c_{j,2} + c_{j,2}^{-1} c_{j,1}).
\end{aligned}
$$

On étudie maintenant les valeurs de $Q^1(j, \mathcal{P}, \mathbf{k})$ :

• Si $\mathcal{P} = \emptyset$ et $\mathbf{k} = (0,0)$, on a

$$
Q^1(j, \emptyset, (0,0)) = \sum_{\substack{f=1 \\ f \neq j}}^{Q} F_{\emptyset, j, f, \mathbf{0}}(\mathbf{0}, \mathbf{0}) + (Q-1)\gamma.
$$

D'après l'Exemple 1.3.4, on a que $F_{\emptyset,j,f,\mathbf{0}}(\mathbf{0},\mathbf{0})) = 0$ donc $Q^1(j,\emptyset, (0,0)) = (Q-1)\gamma$.
* Si $\mathcal{P} = \{1\}$ et $k_2 = 1$, on a

$$ 
\begin{aligned}
Q^1(j, \{1\}, 1) &= (Q-1)\gamma c_{j,2} + \sum_{\substack{f=1 \\ f \neq j}}^{Q} \sum_{w'=0}^{1} F_{\{1,j,f,0,w'e_f+(1-w')e_j(0,0)\}} \left( \binom{1}{w'} c_{f,2}^{w'} c_{j,2}^{1-w'} \right) \\
&= (Q-1)\gamma c_{j,2} + \sum_{\substack{f=1 \\ f \neq j}}^{Q} \left( F_{\{1,j,f,0,e_f(0,0)\}} c_{f,2} + F_{\{1,j,f,0,e_j(0,0)\}} c_{j,2} \right).
\end{aligned}
$$

On trouve à l'aide de l'Exemple 1.3.6 que

$$ F_{\{1,j,f,0,e_j(0,0)\}} = -\frac{1}{c_{j,1}} \ln \left( 1 + \frac{c_{f,1}}{c_{j,1}} \right), $$

et à l'aide de l'Exemple 1.3.5, on a que

$$ F_{\{1,j,f,0,e_f(0,0)\}} = \frac{1}{c_{f,1}} \ln \left( 1 + \frac{c_{f,1}}{c_{j,1}} \right), $$

On obtient alors

$$ Q^1(j, \{1\}, 1) = (Q-1)\gamma c_{j,2} + \sum_{\substack{f=1 \\ f \neq j}}^{Q} \left( \frac{c_{f,2}}{c_{f,1}} - \frac{c_{j,2}}{c_{j,1}} \right) \ln \left( 1 + \frac{c_{f,1}}{c_{j,1}} \right). $$

* Si $\mathcal{P} = \{1\}$ et $k_1 = 1$, mutatis mutandis, on trouve

$$ Q^1(j, \{2\}, 1) = (Q-1)\gamma c_{j,1} + \sum_{\substack{f=1 \\ f \neq j}}^{Q} \left( \frac{c_{f,1}}{c_{f,2}} - \frac{c_{j,1}}{c_{j,2}} \right) \ln \left( 1 + \frac{c_{f,2}}{c_{j,2}} \right). $$

Des trois points précédents, on obtient

$$ 
\begin{aligned}
Z'(-(\mathbf{0}, \mathbf{0})) = & \\
& Q\zeta(0)^2\gamma - \frac{2}{Q+1}\zeta(-1) \left( \sum_{j=1}^{Q} (c_{j,1} + c_{j,2}) \right) \gamma - \frac{Q-1}{Q+1}\zeta(-1) \left( \sum_{j=1}^{Q} (c_{j,1} + c_{j,2}) \right) \gamma \\
& - \frac{1}{Q+1}\zeta(-1) \sum_{\substack{j=1 \\ f \neq j}}^{Q} \left( \left( \frac{c_{f,1}}{c_{f,2}} - \frac{c_{j,1}}{c_{j,2}} \right) \ln \left( 1 + \frac{c_{f,2}}{c_{j,2}} \right) + \left( \frac{c_{f,2}}{c_{f,1}} - \frac{c_{j,2}}{c_{j,1}} \right) \ln \left( 1 + \frac{c_{f,1}}{c_{j,1}} \right) \right) \\
& + \frac{\zeta'(0)\zeta(0)}{Q} - \frac{\zeta'(-1)}{Q+1} \sum_{j=1}^{Q} (c_{j,1} + c_{j,2}) - \gamma\zeta(0)^2 \sum_{j=1}^{Q} (c_{j,1}c_{j,2})^{-1} + \gamma\zeta(-1) \sum_{j=1}^{Q} (c_{j,1}^{-1}c_{j,2} + c_{j,2}^{-1}c_{j,1}) \\
& + \sum_{j=1}^{Q} \partial_s \zeta^B(s, d'_j|c_{j,1}, c_{j,2})|_{s=0}.
\end{aligned}
$$

□

**Remarque 6.3.3.** Avec les hypothèses du lemme précédent, et en fournissant l'entier $Q \ge 1$ et les
coefficients $c = (c_{q,p})_{q \in [1,Q], p \in [1,2]}$, le code SAGE 7.1 permet de simplifier l'expression

$$
\begin{align*}
& Q\zeta(0)^2\gamma - \frac{Q-1}{Q+1}\zeta(-1)\left(\sum_{j=1}^{Q}(c_{j,1} + c_{j,2})\right)\gamma \\
& \quad - \frac{1}{Q+1}\zeta(-1)\sum_{j=1}^{Q}\sum_{\substack{f=1 \\ f \neq j}}^{Q}\left(\left(\frac{c_{f,1}}{c_{f,2}} - \frac{c_{j,1}}{c_{j,2}}\right)\ln\left(1 + \frac{c_{f,2}}{c_{j,2}}\right) + \left(\frac{c_{f,2}}{c_{f,1}} - \frac{c_{j,2}}{c_{j,1}}\right)\ln\left(1 + \frac{c_{f,1}}{c_{j,1}}\right)\right) \\
& \quad + \frac{\zeta'(0)\zeta(0)}{Q} - \frac{\zeta'(-1)}{Q+1}\sum_{j=1}^{Q}(c_{j,1} + c_{j,2}) - \gamma\zeta(0)^2\sum_{j=1}^{Q}(c_{j,1}c_{j,2})^{-1} + \gamma\zeta(-1)\sum_{j=1}^{Q}(c_{j,1}^{-1}c_{j,2} + c_{j,2}^{-1}c_{j,1}).
\end{align*}
$$

Notons au passage que le résultat du code n’est pas une approximation, mais s’exprime en fonction de la constante d’Euler $\gamma$, $\zeta(0)$, $\zeta(-1)$, $\zeta'(0)$, $\zeta'(-1)$ et de certains logarithmes d’entiers.

Même si on dispose d'une formule pour calculer les valeurs aux entiers négatifs de la dérivée suivant s
de la fonction zêta de Barnes $\partial_s \zeta^B(s, d'_j|c_{j,1}, c_{j,2})|_{s=0}$ via le Corollaire B1, en posant $N = 0$ et $\mathbf{R} = \mathbf{0}$, on
utilisera plutôt le Théorème 1.2.11 démontré par Aoki et Sakane. La formule de ce théorème donne des
formules plus facilement exploitable. En particulier, on utilisera les formules obtenues dans l'Exemple
1.2.12 pour simplifier les expressions de $\zeta'_{\mathfrak{g}_2}(0)$ et de $\zeta'_{\mathfrak{g}_0(5)}(0)$. On simplifiera des termes de cet exemple
via les formules (1.4), (1.3) et (1.2).

$$
\begin{align}
\zeta^{B'}(0, 2|1, 1) &= -\zeta'(0) + \zeta'(-1), \tag{6.3} \\
&= \frac{1}{2}\ln(2\pi) + \frac{1}{12} - \ln(A) \nonumber
\end{align}
$$

$$
\begin{align}
\zeta^{B'}(0, 3|1, 2) &= -\frac{5}{8}\ln(2) - \frac{1}{4}\ln(\pi) - \frac{3}{2}\zeta'(0) + \frac{1}{2}\zeta'(-1), \tag{6.4} \\
&= \frac{1}{8}\ln(2) + \frac{1}{2}\ln(\pi) + \frac{1}{24} - \frac{\ln(A)}{2} \nonumber
\end{align}
$$

$$
\zeta^{B'}(0, 4|1, 3) = - \frac{19 \ln(3)}{36} - \zeta'(0) - \frac{2}{3} \zeta' \left(0, \frac{5}{3}\right) - \frac{1}{3} \zeta' \left(0, \frac{4}{3}\right) + \zeta'(-1) + \zeta' \left(-1, \frac{5}{3}\right) + \zeta' \left(-1, \frac{4}{3}\right), \quad (6.5)
$$

$$
\begin{equation}
\begin{split}
\zeta^{B'}(0, 5|2, 3) = & -\frac{31 \ln(6)}{72} - \frac{\ln(2)}{6} - \frac{1}{4} \ln(\pi) - \frac{3}{2} \zeta'(0) - \frac{2}{3} \zeta'\left(0, \frac{5}{3}\right) - \frac{1}{6} \zeta'\left(0, \frac{7}{6}\right) + \frac{1}{6} \zeta'\left(0, \frac{5}{6}\right) \\
& \qquad + \frac{1}{2} \zeta'(-1) + \zeta'\left(-1, \frac{5}{3}\right) + \zeta'\left(-1, \frac{4}{3}\right) + \zeta'\left(-1, \frac{7}{6}\right) + \zeta'\left(-1, \frac{5}{6}\right),
\end{split}
\tag{6.6}
\end{equation}
$$

on a notamment utilisé le fait que

$$
\begin{align*}
\zeta' \left(0, \frac{3}{2}\right) &= \zeta'(0) - \ln(2) + \ln \Gamma \left(\frac{1}{2}\right), \\
&= \zeta'(0) + \ln \left(\sqrt{\pi}\right) - \ln(2),
\end{align*}
$$

et que

$$
\begin{align*}
\zeta' \left(-1, \frac{3}{2}\right) &= \zeta' \left(-1, \frac{1}{2}\right) - \frac{1}{2} \ln(2) \\
&= -\frac{\ln(2)}{6} - \frac{1}{2} \zeta'(-1) - \frac{1}{2} \ln(2) \\
&= -\frac{2}{3} \ln(2) - \frac{1}{2} \zeta'(-1).
\end{align*}
$$
### 6.3.1 Pôles et résidus de $\zeta_{g_2}$

Via [KMT11a], on a vu en 1.6 que $\zeta_{g_2}$ a des pôles de la forme $s = \frac{1}{3}$ et en $s = \frac{1-k}{5}$, avec $k \ge 0$ et $k \ne 1$ mod (5). On souhaite ici donner une expression des résidus de $\zeta_{g_2}$ en $s = \frac{1}{5}$ et en $s = \frac{1}{3}$. On pose

$$ 
\begin{aligned}
l_1(x_1, x_2) &= x_1 + x_2, \quad l_2(x_1, x_2) = x_1 + 2x_2, \quad l_3(x_1, x_2) = x_1 + 3x_2, \quad l_4(x_1, x_2) = 2x_1 + 3x_2 \\
d_1 &= 1, \quad d_2 = 1, \quad d'_1 = 2, \quad d'_2 = 3, \quad d'_3 = 4, \quad d'_4 = 5.
\end{aligned}
$$

Via la Proposition 2.2.4, on trouve que

$$ \zeta_{g_2}(s) = K(\theta, s) + J(\theta, s), $$

avec

$$ K(\theta, s) = 120^{-s} \sum_{\emptyset \neq Q \subset [1,4]} \sum_{A_Q \subset Q^c} (-1)^{|Q^c| - |A_Q|} \frac{1}{\Gamma(s)^{4 - |A_Q|}} \left( \sum_{\mathbf{n} \in \mathbb{N}^2} (n_1 + 1)^{-s} (n_2 + 1)^{-s} \cdot \prod_{q \in A_Q} (l_q(\mathbf{n} + \mathbf{1}))^{-s} \prod_{q \in [1,4] \setminus A_Q} \Gamma(s, \theta, l_q(\mathbf{n} + \mathbf{1})) \right), $$

et

$$ 
\begin{aligned}
J(\theta, s) = & 120^s \sum_{\mathcal{P} \subset [1,2]} \Gamma(1-s)^{|\mathcal{P}|} \sum_{\mathbf{k}=(k_p)_{p \in \mathcal{P}} \in \mathbb{N}^{\mathcal{P}}} (-1)^{|\mathbf{k}|} \left( \sum_{j=1}^4 h_{\mathcal{P},j,\mathbf{k}}(s) \right) \frac{\theta^{(4+|\mathcal{P}|)s-|\mathcal{P}|+|\mathbf{k}|}}{\Gamma(s)((4+|\mathcal{P}|)s-|\mathcal{P}|+|\mathbf{k}|)} \prod_{p \in \mathcal{P}^c} \frac{\zeta(s-k_p)}{k_p!} \\
& + 120^s \Gamma(1-s)^2 \left( \sum_{j=1}^4 h_{[1,2],j}(s) \right) \frac{\theta^{6s-2}}{\Gamma(s)(6s-2)},
\end{aligned}
\quad (6.7) $$

avec

$$ h_{\mathcal{P},1,\mathbf{k}}(s) = \frac{1}{\Gamma(s)^3} \int_{[0,1]^3} (x_2 x_3 x_4)^{s-1} \prod_{p \in \mathcal{P}} l_p^*(1, x_2, x_3, x_4)^{s-1} \prod_{p \in \mathcal{P}^c} l_p^*(1, x_2, x_3, x_4)^{k_p} dx_2 dx_3 dx_4, $$

$$ h_{\mathcal{P},2,\mathbf{k}}(s) = \frac{1}{\Gamma(s)^3} \int_{[0,1]^3} (x_1 x_3 x_4)^{s-1} \prod_{p \in \mathcal{P}} l_p^*(x_1, 1, x_3, x_4)^{s-1} \prod_{p \in \mathcal{P}^c} l_p^*(x_1, 1, x_3, x_4)^{k_p} dx_1 dx_3 dx_4, $$

$$ h_{\mathcal{P},3,\mathbf{k}}(s) = \frac{1}{\Gamma(s)^3} \int_{[0,1]^3} (x_1 x_2 x_4)^{s-1} \prod_{p \in \mathcal{P}} l_p^*(x_1, x_2, 1, x_4)^{s-1} \prod_{p \in \mathcal{P}^c} l_p^*(x_1, x_2, 1, x_4)^{k_p} dx_1 dx_2 dx_4, $$

$$ h_{\mathcal{P},4,\mathbf{k}}(s) = \frac{1}{\Gamma(s)^3} \int_{[0,1]^3} (x_1 x_2 x_3)^{s-1} \prod_{p \in \mathcal{P}} l_p^*(x_1, x_2, x_3, 1)^{s-1} \prod_{p \in \mathcal{P}^c} l_p^*(x_1, x_2, x_3, 1)^{k_p} dx_1 dx_2 dx_3, $$

$$ l_1^*(x_1, x_2) = x_1 + x_2 + x_3 + 2x_4, $$

$$ l_2^*(x_1, x_2) = x_1 + 2x_2 + 3x_3 + 4x_4, $$
et où on a également

$$
\begin{align*}
\sum_{j=1}^{4} h_{[[1,2],j}(s) ={}& \frac{1}{\Gamma(s)^3} \int_{[0,1]^3} (x_1 x_2 x_3)^{s-1} \prod_{p=1}^{2} l_p^*(x_1, x_2, x_3, 1)^{s-1} dx_1 dx_2 dx_3, \\
& + \frac{1}{\Gamma(s)^3} \int_{[0,1]^3} (x_1 x_2 x_4)^{s-1} \prod_{p=1}^{2} l_p^*(x_1, x_2, 1, x_4)^{s-1} dx_1 dx_2 dx_4 \\
& + \frac{1}{\Gamma(s)^3} \int_{[0,1]^3} (x_1 x_3 x_4)^{s-1} \prod_{p=1}^{2} l_p^*(x_1, 1, x_3, x_4)^{s-1} dx_1 dx_3 dx_4 \\
& + \frac{1}{\Gamma(s)^3} \int_{[0,1]^3} (x_2 x_3 x_4)^{s-1} \prod_{p=1}^{2} l_p^*(1, x_2, x_3, x_4)^{s-1} dx_2 dx_3 dx_4.
\end{align*}
$$

On observe que $K(\theta, s)$ est holomorphe sur $\mathbb{C}$, donc ne contribue pas aux résidus. Il ne reste qu’à étudier les dénominateurs des termes présents dans la formule 6.7.

* Soit $\mathcal{P} \subset [[1,2]]$, et $\mathbf{k} = (k_p)_{p \in \mathcal{P}^c} \in \mathbb{N}^{\mathcal{P}^c}$. Pour $s = \frac{1}{5}$, on a

$$ (4 + \mathcal{P})s - |\mathcal{P}| + |\mathbf{k}| = 0 \text{ si et seulement si } |\mathcal{P}| = 1, \mathbf{k} = 0. $$

On trouve alors

$$
\begin{align*}
\mathrm{Res}_{s=\frac{1}{5}}(\zeta_{\mathfrak{g}_2}(s)) &= 120^{\frac{1}{5}} \frac{\Gamma(\frac{4}{5})}{\Gamma(\frac{1}{5})} \left( \sum_{j=1}^{4} h_{\{1\},j,0} \left(\frac{1}{5}\right) + h_{\{2\},j,0} \left(\frac{1}{5}\right) \right) \zeta \left(\frac{1}{5}\right), \\
&= 120^{\frac{1}{5}} \frac{\Gamma(\frac{4}{5})}{\Gamma(\frac{1}{5})^4} \zeta \left(\frac{1}{5}\right) \left( \int_{[0,1]^3} (x_2 x_3 x_4)^{-\frac{4}{5}} (l_1^*(1, x_2, x_3, x_4)^{-\frac{4}{5}} + l_2^*(1, x_2, x_3, x_4)^{-\frac{4}{5}}) dx_2 dx_3 dx_4 \right. \\
&\qquad \left. + \int_{[0,1]^3} (x_1 x_3 x_4)^{-\frac{4}{5}} (l_1^*(x_1, 1, x_3, x_4)^{-\frac{4}{5}} + l_2^*(x_1, 1, x_3, x_4)^{-\frac{4}{5}}) dx_1 dx_3 dx_4 \right. \\
&\qquad \left. + \int_{[0,1]^3} (x_1 x_2 x_4)^{-\frac{4}{5}} (l_1^*(x_1, x_2, 1, x_4)^{-\frac{4}{5}} + l_2^*(x_1, x_2, 1, x_4)^{-\frac{4}{5}}) dx_1 dx_2 dx_4 \right. \\
&\qquad \left. + \int_{[0,1]^3} (x_1 x_2 x_3)^{-\frac{4}{5}} (l_1^*(x_1, x_2, x_3, 1)^{-\frac{4}{5}} + l_2^*(x_1, x_2, x_3, 1)^{-\frac{4}{5}}) dx_1 dx_2 dx_3 \right).
\end{align*}
$$

* Via un raisonnement similaire, on trouve

$$
\begin{align*}
\mathrm{Res}_{s=\frac{1}{3}}(\zeta_{\mathfrak{g}_2}(s)) &= 120^{\frac{1}{3}} \frac{\Gamma(\frac{2}{3})^2}{\Gamma(\frac{1}{3})^4} \left( \sum_{j=1}^{4} h_{\{1,2\},j} \left(\frac{1}{3}\right) + h_{\{1,2\},j} \left(\frac{1}{5}\right) \right) \zeta \left(\frac{1}{3}\right), \\
&= 120^{\frac{1}{3}} \frac{\Gamma(\frac{2}{3})^2}{\Gamma(\frac{1}{3})} \left( \int_{[0,1]^3} (x_1 x_2 x_3)^{-\frac{2}{3}} \prod_{p=1}^{2} l_p^*(x_1, x_2, x_3, 1)^{-\frac{2}{3}} dx_1 dx_2 dx_3 \right. \\
&\qquad \left. + \int_{[0,1]^3} (x_1 x_2 x_4)^{-\frac{2}{3}} \prod_{p=1}^{2} l_p^*(x_1, x_2, 1, x_4)^{-\frac{2}{3}} dx_1 dx_2 dx_4 \right. \\
&\qquad \left. + \int_{[0,1]^3} (x_1 x_3 x_4)^{-\frac{2}{3}} \prod_{p=1}^{2} l_p^*(x_1, 1, x_3, x_4)^{-\frac{2}{3}} dx_1 dx_3 dx_4 \right. \\
&\qquad \left. + \int_{[0,1]^3} (x_2 x_3 x_4)^{-\frac{2}{3}} \prod_{p=1}^{2} l_p^*(1, x_2, x_3, x_4)^{-\frac{2}{3}} dx_2 dx_3 dx_4 \right).
\end{align*}
$$
### 6.3.2 Valeurs aux entiers négatifs de $\zeta_{\mathfrak{g}_2}$ et de $\zeta'_{\mathfrak{g}_2}$

On reprend les notations de la sous-section précédente :

$$l_1(x_1, x_2) = x_1 + x_2, \quad l_2(x_1, x_2) = x_1 + 2x_2, \quad l_3(x_1, x_2) = x_1 + 3x_2, \quad l_4(x_1, x_2) = 2x_1 + 3x_2 \\ d_1 = 1, \quad d_2 = 1, \quad d'_1 = 2, \quad d'_2 = 3, \quad d'_3 = 4, \quad d'_4 = 5.$$

La fonction zêta $\zeta_{\mathfrak{g}_2}$ est de la forme

$$\zeta_{\mathfrak{g}_2}(s) = 120^s \sum_{n_1, n_2 \ge 1} \frac{1}{n_1^{s_1} n_2^{s_2} (n_1 + n_2)^{s_1} (n_1 + 2n_2)^{s_2} (n_1 + 3n_3)^{s_3} (2n_1 + 3n_2)^{s_4}} = 120^s Z_{\Delta}(s).$$

On a en particulier que

$$\zeta_{\mathfrak{g}_2}(-N) = 120^{-N} Z(-(N, N), (N, N, N, N)), \quad \zeta'_{\mathfrak{g}_2}(-N) = 120^{-N} Z'(-(N, N), (N, N, N, N)) + \ln(120) \zeta_{\mathfrak{g}_2}(-N). \\ (1,1), (1,1,1,1) \qquad (1,1), (1,1,1,1)$$

Via le Corollaire A1, on obtient alors

$$\zeta_{\mathfrak{g}_2}(0) = \sum_{\mathcal{P} \subsetneq [1,2]} \frac{(-1)^{|\mathcal{P}|}}{4 + |\mathcal{P}|} \sum_{\mathbf{k}=(k_p)_{p \in \mathcal{P}^c} \in \mathbb{N}^{\mathcal{P}^c} \atop |\mathbf{k}|=|\mathcal{P}|} \left( \sum_{j=1}^4 \prod_{p \in \mathcal{P}^c} c_{j,p}^{k_p} \right) \frac{\zeta(-k_p)}{k_p!}.$$

1) Si $\mathcal{P} = \emptyset$ et $|\mathbf{k}| = 0$, alors $\mathbf{k} = (0,0)$, et

$$\sum_{j=1}^{4} \prod_{p=1}^{2} c_{j,p}^{0} = 4.$$

2) Si $\mathcal{P} = \{1\}$ et $|\mathbf{k}| = 1$, alors $\mathbf{k} = k_2 = 1$, et

$$\sum_{j=1}^{4} c_{j,2}^{1} = 9.$$

3) Si $\mathcal{P} = \{2\}$ et $|\mathbf{k}| = 1$, alors $\mathbf{k} = k_1 = 1$, et

$$\sum_{j=1}^{4} c_{j,1}^{1} = 5.$$

En utilisant le fait que $\zeta(0) = -\frac{1}{2}$, et que $\zeta(-1) = -\frac{1}{12}$, on obtient alors la valeur de $\zeta_{\mathfrak{g}_2}$ en $s = 0$ :

$$\zeta_{\mathfrak{g}_2}(0) = \zeta(0)^2 - \frac{14}{5} \zeta(-1) \\ = \frac{29}{60}.$$

On peut également fournir une expression des valeurs aux entiers négatifs en général. Par le Corollaire A2, on a

$$\zeta_{\mathfrak{g}_2}(-N) = 120^{-N} \sum_{\mathcal{P} \subsetneq [1,2]} \frac{(-1)^{(|\mathcal{P}|+5)N+|\mathcal{P}|}}{4 + |\mathcal{P}|} N!^{|\mathcal{P}|+1} \sum_{\substack{\mathbf{k}=(k_p)_{p \in \mathcal{P}^c} \in \mathbb{N}^{\mathcal{P}^c} \\ |\mathbf{k}|=(4+|\mathcal{P}|)N+|\mathcal{P}|}} Q_N^0(\mathcal{P}, \mathbf{k}) \prod_{p \in \mathcal{P}^c} \frac{\zeta(-N_p - k_p)}{k_p!},$$
avec

$$
Q_N^0(\mathcal{P}, \mathbf{k}) = (-1)^{3N} N!^3 \sum_{\substack{j=1 \\ \mathbf{w}=(\mathbf{w}_p)_{p \in \mathcal{P}^c} \in (\mathbb{N}^4)^{\mathcal{P}^c} \\ \forall p \in \mathcal{P}^c | \mathbf{w}_p|=k_p}} \sum_{\substack{\mathbf{v}=(\mathbf{v}_p)_{p \in \mathcal{P}} \in (\mathbb{N}^{[1,4]\setminus\{j\}})^{\mathcal{P}} \\ \forall q \neq j, \mathbf{v}(q)+\mathbf{w}(q)=N}} \sum_{\substack{\mathbf{p} \in \mathcal{P}^c \\ \prod_{p \in \mathcal{P}^c} c_{q,p}^{w_p,q}}} \left( \prod_{p \in \mathcal{P}^c} \binom{k_p}{\mathbf{w}_p} \prod_{q=1}^4 c_{q,p}^{w_p,q} \right) \left( \prod_{p \in \mathcal{P}} \binom{-N-1}{|\mathbf{v}_p|} \binom{|\mathbf{v}_p|}{\mathbf{v}_p} \prod_{\substack{q=1 \\ q \neq j}}^4 c_{q,p}^{v_p,q} \right).
$$

Pour calculer $\zeta'_{g_2}(0)$, on remarque que

$$
\zeta'_{g_2}(0) = \ln(120)\zeta_{g_2}(0) + Z'_{\Delta}(0).
$$

En appliquant le Lemme 6.3.2, et en utilisant le code SAGE 7.1 on obtient

$$
\zeta'_{g_2}(0) = \ln(120)\zeta_{g_2}(0) + 2\gamma\zeta(0)^2 + \frac{1}{4}\zeta(0)\zeta'(0) - 4\gamma\zeta(-1) + \frac{1}{5}\zeta(-1)(\ln(3) + 2\ln(2)) - \frac{14}{5}\zeta'(-1) \\
+ \zeta^{B'}(0, 2|1, 1) + \zeta^{B'}(0, 3|1, 2) + \zeta^{B'}(0, 4|1, 3) + \zeta^{B'}(0, 5|2, 3).
$$

Par les formules 6.3, 6.4, 6.5 et 6.6, on trouve que

$$
\begin{align*}
\zeta^{B'}(0, 2|1, 1) + \zeta^{B'}(0, 3|1, 2) + \zeta^{B'}(0, 4|1, 3) + \zeta^{B'}(0, 5|2, 3) = & \\
& - \zeta'(0) + \zeta'(-1) - \frac{5}{8}\ln(2) - \frac{1}{4}\ln(\pi) - \frac{3}{2}\zeta'(0) + \frac{1}{2}\zeta'(-1) \\
& - \frac{19\ln(3)}{36} - \zeta'(0) - \frac{2}{3}\zeta'\left(0, \frac{5}{3}\right) - \frac{1}{3}\zeta'\left(0, \frac{4}{3}\right) + \zeta'(-1) + \zeta'\left(-1, \frac{5}{3}\right) + \zeta'\left(-1, \frac{4}{3}\right) \\
& - \frac{31\ln(6)}{72} - \frac{\ln(2)}{6} - \frac{1}{4}\ln(\pi) - \frac{3}{2}\zeta'(0) - \frac{2}{3}\zeta'\left(0, \frac{5}{3}\right) - \frac{1}{6}\zeta'\left(0, \frac{7}{6}\right) + \frac{1}{6}\zeta'\left(0, \frac{5}{6}\right) \\
& + \frac{1}{2}\zeta'(-1) + \zeta'\left(-1, \frac{5}{3}\right) + \zeta'\left(-1, \frac{4}{3}\right) + \zeta'\left(-1, \frac{7}{6}\right) + \zeta'\left(-1, \frac{5}{6}\right).
\end{align*}
$$

En simplifiant ces termes, on trouve :

$$
\begin{align*}
\zeta'_{g_2}(0) ={}& \ln(120)\zeta_{g_2}(0) + 2\gamma\zeta(0)^2 + \frac{1}{4}\zeta(0)\zeta'(0) - 4\gamma\zeta(-1) + \frac{1}{5}\zeta(-1)(\ln(3) + 2\ln(2)) - \frac{14}{5}\zeta'(-1) \\
& - \frac{11}{9}\ln(2) - \frac{23}{24}\ln(3) - \frac{1}{2}\ln(\pi) - 5\zeta'(0) + 3\zeta'(-1) - \frac{4}{3}\zeta'\left(0, \frac{5}{3}\right) - \frac{1}{3}\zeta'\left(0, \frac{4}{3}\right) \\
& + \zeta'\left(-1, \frac{5}{3}\right) + \zeta'\left(-1, \frac{4}{3}\right) - \frac{1}{6}\zeta'\left(0, \frac{7}{6}\right) + \frac{1}{6}\zeta'\left(0, \frac{5}{6}\right).
\end{align*}
$$

6.3.3 Pôles et résidus de ζ<sub>SO(5)</sub>

Via [KMT10d], on a vu en 1.5 que $\zeta_{so(5)}$ a des pôles de la forme $s = \frac{1}{2}$ et en $s = \frac{1-k}{3}$, pour $k \ge 0$
un entier quelconque tel que $k \ne 1$ mod (3). On se propose ici d'obtenir des expressions des résidus en
$s = \frac{1}{3}$ et en $s = \frac{1}{2}$ pour $\zeta_{so(5)}(s)$. Notons que Bridges, Brindle, Bringmann, et Franke ont déjà obtenu une
expression très simple de ces deux résidus dans [BBBF23, Proposition 5.16] via des techniques différentes.

On sait que, pour $\theta > 0$ suffisamment petit, la fonction zêta $\zeta_{s\theta(5)}$ admet la décomposition $\zeta_{s\theta(5)}(s) = K(\theta, s) + J(\theta, s)$, avec $K(\theta, s)$ une fonction holomorphe en $s$ sur tout $\mathbb{C}$, et $J(\theta, s)$ une fonction méromorphe
en $s$ sur tout $\mathbb{C}$, ayant les mêmes pôles que $\zeta_{s_0(5)}(s)$. Par la Proposition 2.2.4, on a

$$
\begin{equation} \tag{6.8}
\begin{split}
J(\theta, s) = {}& 6^s \sum_{\mathcal{P} \subsetneq [1,2]} \Gamma(1-s)^{|\mathcal{P}|} \sum_{\mathbf{k}=(k_p)_{p \in \mathcal{P}} \in \mathbb{N}^{\mathcal{P}^c}} (-1)^{|\mathbf{k}|} \left( \sum_{j=1}^2 h_{\mathcal{P},j,\mathbf{k}}(s) \right) \frac{\theta^{(2+|\mathcal{P}|)s-|\mathcal{P}|+|\mathbf{k}|}}{\Gamma(s)((2+|\mathcal{P}|)s-|\mathcal{P}|+|\mathbf{k}|)} \prod_{p \in \mathcal{P}^c} \frac{\zeta(s-k_p)}{k_p!} \\
& + 6^s \Gamma(1-s)^2 \left( \sum_{j=1}^2 h_{[1,2],j,0}(s) \right) \frac{\theta^{4s-2}}{\Gamma(s)(4s-2)},
\end{split}
\end{equation}
$$

avec

$$
h_{\mathcal{P},1,\mathbf{k}}(s) = \frac{1}{\Gamma(s)} \int_0^1 x_2^{s-1} \prod_{p \in \mathcal{P}} l_p^*(1, x_2)^{s-1} \prod_{p \in \mathcal{P}^c} l_p^*(x_1, x_2)^{k_p} dx_2,
$$

$$
h_{\mathcal{P},2,\mathbf{k}}(s) = \frac{1}{\Gamma(s)} \int_0^1 x_1^{s-1} \prod_{p \in \mathcal{P}} l_p^*(x_1, 1)^{s-1} \prod_{p \in \mathcal{P}^c} l_p^*(x_1, 1)^{k_p} dx_1,
$$

$$
l_1^*(x_1, x_2) = x_1 + x_2,
$$

$$
l_2^*(x_1, x_2) = x_1 + 2x_2.
$$

Comme $K(\theta, s)$ est holomorphe en $s$ sur $\mathbb{C}$, on devrait pouvoir lire les résidus de $\zeta_{s_0(5)}(s)$ sur l'expression de $J(\theta, s)$ ci-dessus. En $s = 1/2$, le seul terme contribuant au résidu de $\zeta_{s_0(5)}(s)$ est

$$
\Gamma(1-s)^2 \left( \sum_{j=1}^{2} h_{[1,2],j,0}(s) \right) \frac{\theta^{4s-2}}{\Gamma(s)(4s-2)}. \text{ On obtient alors une expression du résidu en } s = 1/2 :
$$

$$
\begin{align*}
\mathrm{Res}_{s=1/2}(\zeta_{s0}(5)(s)) &= \sqrt{6} \frac{\Gamma(3/2)^2}{4\Gamma(1/2)} \left( \sum_{j=1}^2 h_{[1,2],j,0}(1/2) \right), \\
&= \sqrt{6} \frac{\Gamma(3/2)^2}{4\Gamma(1/2)} \frac{1}{\Gamma(1/2)} \left( \int_0^1 x^{-1/2}(x+1)^{-1/2}(x+2)^{-1/2} dx + \int_0^1 x^{-1/2}(1+x)^{-1/2}(1+2x)^{-1/2} dx \right) \\
&= \sqrt{6} \frac{\Gamma(3/2)^2}{4\Gamma(1/2)} \frac{1}{\Gamma(1/2)} \left( \int_0^1 x^{-1/2}(x+1)^{-1/2}(x+2)^{-1/2} dx + \int_0^1 x^{-1/2}(1+x)^{-1/2}(1+2x)^{-1/2} dx \right).
\end{align*}
$$

Pour les pôles de la forme $\frac{n}{3}$, avec $n \le 1$ un entier tel que $n \ne 0 (\mathrm{mod}~3)$, les seuls termes contribuant au résidu en $s = \frac{n}{3}$ correspondent aux termes de la forme

$$
-\Gamma(1 - n/3)^{|\mathcal{P}|} (-1)^{|\mathbf{k}|} \left( \sum_{j=1}^{2} h_{\mathcal{P},j,\mathbf{k}}(1 - n/3) \right) \frac{1}{\Gamma(n/3)((2 + |\mathcal{P}|))} \prod_{p \in \mathcal{P}^c} \frac{\zeta(n/3 - k_p)}{k_p!},
$$

avec $\mathcal{P} \subsetneq [1,2], \mathbf{k} \in \mathbb{N}^{\mathcal{P}^c}$ vérifiant la condition $(2 + |\mathcal{P}|)^{n/3} - |\mathcal{P}| + |\mathbf{k}| = 0$. Cette condition implique nécessairement que $\mathcal{P} = \{i\}$ est un singleton, et en notant $\mathcal{P}^c = \{p\}$, on a $|k| = k_p = 1 - n \in \mathbb{N}$. On trouve alors

$$
\mathrm{Res}_{s=n/3}(\zeta_{s0}(5)(s)) = (-1)^{1-n} \Gamma\left(\frac{3-n}{3}\right) \left( \sum_{i=1}^{2} \sum_{j=1}^{2} h_{\{i\},j,\mathbf{k}}\left(1-\frac{n}{3}\right) \right) \frac{1}{3\Gamma\left(\frac{n}{3}\right)} \frac{\zeta\left(\frac{4n}{3}-1\right)}{(1-n)!},
$$

Notons que, en considérant les termes généraux de la série présente dans 6.8 avec $\mathcal{P} = \emptyset$, on remarque
des termes de la forme $\frac{\theta^{2s+|\mathbf{k}|}}{\Gamma(s)(2s+|\mathbf{k}|)}$, avec $\mathbf{k} = (k_1, k_2) \in \mathbb{N}$. Les zéros de $2s + |\mathbf{k}|$ ne constituent pas
des pôles puisque ces pôles sont annulés par les zéros des termes de la forme $\sum_{j=1}^2 h_{\emptyset,j,\mathbf{k}}$. Par exemple,
en $s = -\frac{1}{2}$, on a que le terme $2s + |k|$ est nul si et seulement si $|k| = k_1 + k_2 = 1$, c'est-à-dire lorsque $k \in \{(1,0), (0,1)\}$. Or on remarque que

$$ 
\begin{aligned}
h_{1,\emptyset,(1,0)}(s) + h_{2,\emptyset,(1,0)}(s) + h_{1,\emptyset,(0,1)}(s) + h_{1,\emptyset,(0,1)}(s) &= \frac{1}{\Gamma(s)} \int_0^1 (x^{s-1}(1+2x) + 2x^{s-1}(1+x) + x^{s-1}(2+x))dx \\
&= \frac{5}{\Gamma(s)} \left( \frac{1}{s+1} + \frac{1}{s} \right),
\end{aligned}
$$

et en $s = -\frac{1}{2}$, ce terme s'annule. Cela est cohérent avec la liste des pôles possibles donnée par Komori, Matsumoto et Tsumura dans [KMT10d, Théorème 6.2].

### 6.3.4 Valeurs aux entiers négatifs de $\zeta_{s0(5)}$ et $\zeta'_{s0(5)}$

On pose

$$ 
\begin{aligned}
l_1(x_1, x_2) &= x_1 + x_2, & l_2(x_1, x_2) &= x_1 + 2x_2, \\
d_1 &= 1, & d_2 &= 1, & d'_1 &= 2, & d'_2 &= 3,
\end{aligned}
$$

alors $\zeta_{s0(5)}(s)$ est de la forme

$$ \zeta_{s0(5)}(s) = 6^s \sum_{\mathbf{n} \in \mathbb{N}^2} \frac{1}{n_1^s n_2^s (n_1 + n_2)^s (n_1 + 2n_2)^s} = 6^s Z_{\Delta}(s). $$

On a en particulier que

$$ \zeta_{s0(5)}(-N) = 6^{-N} Z(-((N,N), (N,N)), \quad \zeta'_{s0}(-N) = 120^{-N} Z'(-((N,N), (N,N)) + \ln(6)\zeta_{s0}(-N)). $$

En $N=0$, on trouve par le Corollaire A1 :

$$ 
\begin{aligned}
\zeta_{s0}(5)(0) &= \sum_{\mathcal{P} \subseteq [1,2]} \frac{(-1)^{|\mathcal{P}|}}{2 + |\mathcal{P}|} \sum_{\mathbf{k}=(k_p)_{p \in \mathcal{P}^c} \in \mathbb{N}^{\mathcal{P}^c} \atop |\mathcal{P}|=|\mathbf{k}|} \left( \sum_{j=1}^2 \prod_{p \in \mathcal{P}^c} c_{j,p}^{k_p} \right) \prod_{p \in \mathcal{P}^c} \frac{\zeta(-k_p)}{k_p!} \\
&= \zeta(0)^2 - \frac{5}{3}\zeta(-1) \\
&= \frac{7}{18}.
\end{aligned}
$$

On peut également fournir une expression des valeurs aux entiers négatifs en général. Par le Corollaire A2, on a

$$ \zeta_{s0(5)}(-N) = 6^{-N} \sum_{\mathcal{P} \subseteq [1,2]} \frac{(-1)^{(|\mathcal{P}|+3)N+|\mathcal{P}|}}{3+|\mathcal{P}|} N!^{|\mathcal{P}|+1} \sum_{\substack{\mathbf{k}=(k_p)_{p \in \mathcal{P}^c} \in \mathbb{N}^{\mathcal{P}^c} \\ |\mathbf{k}|=(3+|\mathcal{P}|)N+|\mathcal{P}|}} Q_N^0(\mathcal{P}, \mathbf{k}) \prod_{p \in \mathcal{P}^c} \frac{\zeta(-N-k_p)}{k_p!}, $$

avec

$$ 
\begin{aligned}
Q_N^0(\mathcal{P}, \mathbf{k}) = N! \sum_{j=1}^2 \sum_{\mathbf{w}=(\mathbf{w}_p)_{p \in \mathcal{P}^c} \in (\mathbb{N}^2)^{\mathcal{P}^c} \atop \forall p \in \mathcal{P}^c | \mathbf{w}_p| = k_p} \sum_{\mathbf{v}=(\mathbf{v}_p)_{p \in \mathcal{P}^c} \in (\mathbb{N}^2 \setminus \{j\})^{\mathcal{P}^c} \atop \forall q \neq j, \mathbf{v}(q)+\mathbf{w}(q)=N} \left( \prod_{p \in \mathcal{P}^c} \binom{k_p}{\mathbf{w}_p} \prod_{q=1}^2 c_{q,p}^{w_p,q} \right) \left( \prod_{p \in \mathcal{P}^c} \binom{-N-1}{|\mathbf{v}_p|} \binom{|\mathbf{v}_p|}{|\mathbf{w}_p|} \prod_{\substack{q=1 \\ q \neq j}}^2 c_{q,p}^{v_p,q} \right).
\end{aligned}
$$
Enfin, par le Lemme 6.3.2, en s'aidant du Code SAGE 7.1 on a l'expression suivante de sa dérivée en 0 :

$$
\begin{align*}
\zeta'_{so(5)}(0) ={}& \ln(6)\zeta_{so(5)}(0) + \frac{1}{2}\gamma\zeta(0)^2 + \frac{1}{2}\zeta(0)\zeta'(0) - \frac{1}{2}\gamma\zeta(-1) - \frac{5}{3}\zeta'(-1) + \frac{1}{6}\ln(2) \\
& + \partial_s\zeta^B(s, 2|1, 1)|_{s=0} + \partial_s\zeta^B(s, 3|1, 2)|_{s=0},
\end{align*}
$$

et avec les formules (6.3) et (6.4), on trouve :

$$
\begin{align*}
\zeta'_{so(5)}(0) ={}& \ln(6)\zeta_{so(5)}(0) + \frac{1}{2}\gamma\zeta(0)^2 + \frac{1}{2}\zeta(0)\zeta'(0) - \frac{1}{2}\gamma\zeta(-1) - \frac{5}{3}\zeta'(-1) + \frac{1}{6}\ln(2) \\
& - \zeta'(0) + \zeta'(-1) - \frac{5}{8}\ln(2) - \frac{1}{4}\ln(\pi) - \frac{3}{2}\zeta'(0) + \frac{1}{2}\zeta'(-1).
\end{align*}
$$

En simplifiant ces termes, on trouve

$$
\zeta'_{so(5)}(0) = \frac{7}{18} \ln(6) - \frac{11}{24} \ln(2) - \frac{1}{4} \ln(\pi) + \left( \frac{1}{2} \zeta(0)^2 - \frac{1}{2} \zeta(-1) \right) \gamma - \frac{11}{4} \zeta'(0) - \frac{13}{6} \zeta'(-1).
$$

L'expression ci-dessus nous permet d'obtenir une expression explicite du coefficient $C$ dans le Théorème 1.2.35.

6.4 Application à l'étude du comportement asymptotique de

$r_{g_2}(n)$

Rappelons que

$$
r_{g_2}(n) = \left| \left\{ (k_{i,j})_{i,j \ge 1} \in (\mathbb{N})^{\mathbb{N}*2} \left| \sum_{i,j \ge 1} k_{i,j} \frac{ij(i+j)(i+2j)(i+3j)(2i+3j)}{5!} = n \right. \right\} \right|.
$$

$$
\text{Posons } P(i, j) := \frac{ij(i+j)(i+2j)(i+3j)(2i+3j)}{5!}. \text{ On observe que l'on a}
$$

$$
\prod_{i,j \ge 1} \frac{1}{1 - q^{P(i,j)}} = \prod_{i,j \ge 1} \sum_{k_{i,j} \ge 1} q^{k_{i,j} P(i,j)},
$$

en développant ce produit infini, et en utilisant l'expression de l'ensemble $r_{g_2}(n)$, on trouve que :

$$
\sum_{n=1}^{+\infty} r_{g_2}(n)q^n = \prod_{i,j \ge 1} \frac{1}{1 - q^{P(i,j)}}.
$$

$$
\text{En posant } f(n) = |\{(i,j) \in \mathbb{N}^{*2} | P(i,j) = n\}|, \text{ on obtient que}
$$

$$
\sum_{n=1}^{+\infty} r_{g_2}(n)q^n = \prod_{n=1}^{+\infty} \frac{1}{(1-q^n)^{f(n)}}.
$$

On note désormais $L_f(s) = \zeta_{g_2}(s)$, et $L_f^*(s) = \zeta_{g_2}(s)\Gamma(s)\zeta(s+1)$. Observons que $L_f^*(s)$ ne possède que deux pôles dans $H_0$, en $s = \frac{1}{3}$ et en $s = \frac{1}{5}$. De plus, $L_f^*(s)$ possède un pôle double en $s = 0$. Remarquons également que, grâce à la liste des pôles obtenues en 1.6, on a que $L_f^*$ possède des pôles simples en $s = \frac{1}{3}$ et en $s = \frac{k}{5}$, $k \in \mathbb{Z}_{\le 1}$, avec $k \ne 0 \bmod 5$.

On souhaite appliquer le Théorème 1.2.33. Vérifions tout d'abord si, avec ces données, les conditions du théorème sont valides. Posons $\Lambda := \mathbb{N} \setminus f^{-1}(\{0\})$. Remarquons que $\frac{3}{2} \cdot \frac{1}{5} < \frac{1}{3} < \frac{2}{1} \cdot \frac{1}{5}$. Ainsi, l'entier $l = 2$ vérifie l'hypothèse du Théorème 1.2.33. Vérifions maintenant si les hypothèses (P1), (P2) et (P3) sont vérifiées :
(P1) Soit $p \ge 2$ un nombre premier. On a que $|\Lambda \setminus (p\mathbb{N} \cap \Lambda)| = +\infty$. En effet, les suites

$$
\begin{align*}
u(2) &:= (P(8k + 1, 1))_{k \in \mathbb{N}}, \\
u(3) &:= (P(9k + 1, 1))_{k \in \mathbb{N}}, \\
u(5) &:= (P(25k + 1, 1))_{k \in \mathbb{N}}, \\
u(p) &:= (P(kp + 1, 1))_{k \ge 1} \quad \text{si } p \ge 7,
\end{align*}
$$

sont strictement croissantes, et appartiennent à Λ par construction. De plus, on observe que la
suite $u(p)$ est incluse dans $\Lambda \setminus (p\mathbb{N} \cap \Lambda)$. On obtient alors que n'importe quel réel $L \ge \frac{1}{6}$ vérifie la
condition (P1).

(P2) On a déjà vu précédemment que $L_f^*$ possède deux pôles (qui sont simples) en $s = \frac{1}{3}$ et en $s = \frac{1}{5}$, un pôle double en $s = 0$, et des pôles simples en $s = \frac{k}{5}$, $k \in \mathbb{Z}_{\le 1}$, avec $k \ne 0 \bmod 5$. Ainsi, pour tout réel $R \in \mathbb{R}_+^*$ tel que $R \ne \frac{k}{5}$ avec $k \ne 0 \bmod 5$, $R$ vérifie la condition de la seconde hypothèse.

(P3) On remarque que le polynôme P vérifie la condition H0S de l'article [Ess97]. En particulier, on sait par [Ess97, Théorème 3] que la fonction zêta rattachée à ce polynôme, (i.e. $\zeta_{\mathfrak{g}_2}$) admet une borne polynomiale en la partie imaginaire de $s = \sigma + i\tau$ sur chaque bande verticale $\sigma_1 \le \sigma \le \sigma_2$. On en déduit alors que la condition (P3) est vérifiée.

**Remarque 6.4.1.** Dans la condition (P2), *R* est arbitrairement grand, et dans la condition (P1), *L* est arbitrairement grand.

Notons $\alpha = \frac{1}{3}$, $\beta = \frac{1}{5}$, $\omega_{\alpha} := \text{Res}_{s=\frac{1}{3}} \zeta_{\mathfrak{g}_2}(s)$ et $\omega_{\beta} := \text{Res}_{s=\frac{1}{5}} \zeta_{\mathfrak{g}_2}(s)$. En appliquant le Théorème 1.2.33, on trouve alors

$$
r_{\mathfrak{g}_2}(n) = \sum_{n \to \infty} \frac{C}{n^b} \exp \left( A_1 n^{\frac{1}{4}} + A_2 n^{\frac{3}{20}} + A_3 n^{\frac{1}{20}} \right) \left( 1 + \sum_{j=2}^{N} \frac{B_j}{n^{\nu_j}} + O_{L,R} \left( n^{-\min\left\{\frac{2L-\alpha}{2(\alpha+1)}, \frac{R}{\alpha+1}\right\}} \right) \right),
$$

avec

$$
C = \frac{e^{\zeta_{\mathfrak{g}_2}^{\prime}(0)} \left(\omega_{\alpha}\Gamma\left(\frac{4}{3}\right)\zeta\left(\frac{4}{3}\right)\right)^{\frac{1-6\zeta_{\mathfrak{g}_2}^{\prime}(0)}{8}} \sqrt{3}}{\sqrt{8\pi}}, \quad b = \frac{6-6\zeta_{\mathfrak{g}_2}(0)+1}{8},
$$

$$
\text{et } A_1 := 4 \left( \omega_\alpha \Gamma \left( \frac{4}{3} \right) \zeta \left( \frac{4}{3} \right) \right)^{\frac{3}{4}}, \quad A_2 := \frac{\omega_\beta \Gamma \left( \frac{1}{5} \right) \zeta \left( \frac{6}{5} \right)}{\left( \omega_\alpha \Gamma \left( \frac{1}{3} \right) \zeta \left( \frac{4}{3} \right) \right)^{\frac{3}{20}}}, \text{ et}
$$

$$
A_3 := K_3 + 3 \left( \omega_\alpha \Gamma \left( \frac{4}{3} \right) \zeta \left( \frac{4}{3} \right) \right)^{\frac{3}{4}} \left( \binom{-1/3}{1} \frac{K_3}{\left( \omega_\alpha \Gamma \left( \frac{4}{3} \right) \zeta \left( \frac{4}{3} \right) \right)^{\frac{3}{4}}} + \binom{-1/3}{2} \frac{K_2^2}{\left( \omega_\alpha \Gamma \left( \frac{4}{3} \right) \zeta \left( \frac{4}{3} \right) \right)^{\frac{3}{2}}} \right) \\
+ \frac{5 \left( \omega_\beta \Gamma \left( \frac{6}{5} \right) \zeta \left( \frac{6}{5} \right) \right)}{\left( \omega_\alpha \Gamma \left( \frac{4}{3} \right) \zeta \left( \frac{4}{3} \right) \right)^{\frac{3}{20}}} \binom{-1/5}{1} \frac{K_2}{\left( \omega_\alpha \Gamma \left( \frac{4}{3} \right) \zeta \left( \frac{4}{3} \right) \right)^{\frac{3}{4}}},
$$

avec $0 < \nu_2 < \dots$ les éléments strictement positifs de $\mathcal{N} + \mathcal{M}$.
Par [BBBF23, Lemme 4.3], on a

$$ K_2 = \frac{3}{4} \cdot \frac{\omega_\beta \Gamma\left(\frac{6}{5}\right) \zeta\left(\frac{6}{5}\right)}{\left(\omega_\alpha \Gamma\left(\frac{4}{3}\right) \zeta\left(\frac{4}{3}\right)\right)^{\frac{3}{20}}} $$ $$ K_3 = - \frac{3}{160} \cdot \frac{\left(\omega_\beta \Gamma\left(\frac{6}{5}\right) \zeta\left(\frac{6}{5}\right)\right)^2}{\left(\omega_\alpha \Gamma\left(\frac{4}{3}\right) \zeta\left(\frac{4}{3}\right)\right)^{\frac{21}{20}}} $$

Des ensembles décrits dans (1.8), (1.9), et (1.10) dans [BBBF23], on trouve que la suite $\nu_j$ du Théorème 1.2.33 donne, avec nos hypothèses, $\nu_2 = \frac{1}{20}, \nu_3 = \frac{2}{20}, \nu_4 = \frac{3}{20}, \dots$
<figure><img src="image_13.png" /></figure>
# Chapitre 7

# Annexe

## 7.1 Codes SAGE

Listing 7.1: Code SAGE pour calculer $Z'_{\Delta}(0)$ avec $P = 2$ et $d_1 = d_2 = 1$

from sage.all import *

c_g2 = [[1,1],[1,2],[1,3],[2,3]]
c.so5 = [[1,1],[1,2]]

gamma = eulergamma
zeta0 = var('zeta0')
zeta_m1 = var('zeta_m1')
zeta0_prime = var('zeta0_prime')
zeta_m1_prime = var('zeta_m1_prime')

def Z_delta_prime_zero(c_matrix):

Q = len(c_matrix)
sum1 = sum(c[j][0] + c[j][1] for j in range(Q))
sum2 = sum(sum((c[f][0]/c[f][1] - c[j][0]/c[j][1])
                 * log(1 + c[f][1]/c[j][1]) + (c[f][1]/c[f][0] - c[j][1]/c[j][0])
                 * log(1 + c[f][0]/c[j][0]))
for f in range(Q) if f != j) for j in range(Q))
sum3 = sum((c[j][0] * c[j][1]) ** -1 for j in range(Q))
sum4 = sum(c[j][0] ** -1 * c[j][1] + c[j][1] ** -1 * c[j][0] for j in range(Q))
result = Q*zeta0 ** 2*gamma - zeta_m1*sum1*gamma - 1/(Q+1)*zeta_m1*sum2
       +zeta0_prime*zeta0/Q - zeta_m1_prime/(Q+1)*sum1 - gamma*zeta0 ** 2*sum3
       + gamma*zeta_m1*sum4
return(result)
print(result)

Listing 7.2: Code SAGE pour calculer $Z_{\Delta}(-N)$ avec $P = 2$

from math import comb
from itertools import product

def alpha_k_N(Q, N, k, c_matrix):
    alpha = 0
    for j in range(1, Q + 1):
        for w in product(range(N + 1), repeat=Q):
if sum(w) == k:
    coef1 = multinomial(w)
    coef2 = multinomial([N - w_i for w_i in w])
prod = 1
for q in range(Q):
    prod *= (c_matrix[q][0] ** w[q]) * (c_matrix[q][1] ** (N - w[q]))
alpha += coef1 * coef2 * prod
return alpha

def beta_N(Q, N, c_matrix):
    beta = 0
    for j in range(1, Q + 1):
        for w in product(range((Q + 1) * N + 2), repeat=Q):
            if sum(w) == (Q + 1) * N + 1 and w[j - 1] >= 2 * N + 1:
                numerator = 1
                for i in range(N + 1 - w[j - 1], -N - 1):
                    numerator *= i

denominator = 1
for q in range(Q):
    if q != j - 1:
        denominator *= factorial(w[q])

term1 = c_matrix[j - 1][1] ** w[j - 1]

term2 = c_matrix[j - 1][0] ** w[j - 1]

for q in range(Q):
    if q != j - 1:
        term1 *= (c_matrix[q][0] ** (N - w[q]))
        term1 *= (c_matrix[q][1] ** w[q])
        term2 *= (c_matrix[q][1] ** (N - w[q]))
        term2 *= (c_matrix[q][0] ** w[q])

beta += (numerator / denominator) * (term1 + term2)
return beta

def Z_Delta_minus_N(N, c_matrix):
    Q = len(c_matrix)
    sum1 = 0
    N_fact_Q = factorial(N) ** Q

for k in range(Q * N + 1):
    alpha = alpha_k_N(Q, N, k, c_matrix)
    term1 = zeta(-N - k) / factorial(k)
    term2 = zeta(-(Q + 1) * N + k) / factorial(Q * N - k)
    sum1 += alpha * term1 * term2

beta = beta_N(Q, N, c_matrix)
sum2 = ((-1) ** (N + 1)) * (N_fact_Q * factorial(N)) * beta
* zeta(-(Q + 2) * N - 1) / factorial((Q + 1) * N + 1)

Z = (1 / Q) * N_fact_Q * sum1 + (1 / (Q + 1)) * sum2

return Z
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Special values of generalized multiple Hurwitz zeta
functions and their derivatives. Applications towards
some Witten zeta functions attached to semi-simple
Lie algebras

PhD in Mathematics
**Simon Rutard**

Under the supervision of Driss Essouabri

Institut Camille Jordan, Université Jean Monnet

January 12, 2024
<figure><img src="image_14.png" /></figure>
# Contents

<table><thead><tr><th></th><th></th><th></th></tr></thead><tbody><tr><td>1</td><td>Introduction</td><td>1</td></tr><tr><td>1.1</td><td>Notations</td><td>1</td></tr><tr><td>1.2</td><td>Some history on zeta and multizeta functions</td><td>3</td></tr><tr><td>1.2.1</td><td>Some results on special values of zeta functions</td><td>4</td></tr><tr><td>1.2.2</td><td>Some results on special values of multizeta functions</td><td>12</td></tr><tr><td>1.2.3</td><td>On asymptotic formulas for some partitions</td><td>14</td></tr><tr><td>1.3</td><td>First definitions</td><td>20</td></tr><tr><td>1.4</td><td>Main results</td><td>23</td></tr><tr><td>2</td><td>Crandall's Expansion of Z(s, s')</td><td>37</td></tr><tr><td>2.1</td><td>Proof of Proposition 1.3.9</td><td>37</td></tr><tr><td>2.2</td><td>Statement of the extension formula for Z<sub>N,N'</sub>(s)</td><td>39</td></tr><tr><td>2.3</td><td>Directional Crandall's expansion for Z(s, s')</td><td>42</td></tr><tr><td>2.3.1</td><td>Lemmata</td><td>42</td></tr><tr><td>2.3.2</td><td>Preliminary Results</td><td>46</td></tr><tr><td>2.3.3</td><td>Proof of Proposition 2.2.2</td><td>57</td></tr><tr><td>2.3.4</td><td>Proof of Proposition 2.2.4</td><td>64</td></tr><tr><td>3</td><td>Computation of the coefficients Q<sup>0</sup><sub>N,N'</sub>(P, j, k) and Q<sup>1</sup><sub>N,N'</sub>(P, j, k)</td><td>69</td></tr><tr><td>3.1</td><td>Lemmata</td><td>69</td></tr><tr><td>3.2</td><td>Value and derivative of h<sub>N,N',P,j,k</sub>(s) at s = 0</td><td>73</td></tr><tr><td>3.2.1</td><td>Preliminaries</td><td>73</td></tr><tr><td>3.2.2</td><td>Computation of Q<sup>0</sup><sub>N,N'</sub>(P, j, k)</td><td>75</td></tr><tr><td>3.2.3</td><td>Computation of Q<sup>1</sup><sub>N,N'</sub>(P, j, k)</td><td>76</td></tr><tr><td>4</td><td>Proof of Theorem A, and of his corollaries</td><td>83</td></tr><tr><td>4.1</td><td>Statement of Theorem A</td><td>83</td></tr><tr><td>4.2</td><td>Proof of Theorem A</td><td>84</td></tr><tr><td>4.3</td><td>On corollaries of Theorem A</td><td>84</td></tr><tr><td>4.3.1</td><td>Proof of Corollary A1</td><td>85</td></tr><tr><td>4.3.2</td><td>Proof of Corollary A2</td><td>85</td></tr><tr><td>4.3.3</td><td>Proof of Corollary A3</td><td>86</td></tr><tr><td>5</td><td>Proof of Proposition B, Theorem D, and some corollaries</td><td>87</td></tr><tr><td>5.1</td><td>Statement of Proposition B and of Theorem D</td><td>87</td></tr><tr><td>5.2</td><td>Auxiliary functions</td><td>89</td></tr><tr><td>5.2.1</td><td>Proof of Proposition B</td><td>92</td></tr><tr><td>5.3</td><td>Proof of Theorem D</td><td>95</td></tr><tr><td>5.4</td><td>On the corollaries of Theorem D</td><td>103</td></tr></tbody></table>
<table><tr><td>5.4.1 Proof of Theorem C</td><td>103</td></tr><tr><td>5.4.2 Proof of Corollary C1</td><td>104</td></tr><tr><td>5.4.3 Proof of Corollary D1</td><td>105</td></tr><tr><td>5.4.4 Proof of Corollary D2</td><td>105</td></tr><tr><td>5.4.5 Proof of Corollary D3</td><td>107</td></tr></table>

6 Witten zeta functions 109

6.1 Generalities on Lie algebras 109

6.2 Root System of a Lie Algebra 110

6.3 Some Results on $\zeta_{g_2}$ and $\zeta_{s_0(5)}$ 113

6.3.1 Poles and residues of $\zeta_{g_2}$ 118

6.3.2 Special values of $\zeta_{g_2}$ and of $\zeta'_{g_2}$ 120

6.3.3 Poles and residues of $\zeta_{s_0(5)}$ 122

6.3.4 Special values of $\zeta_{s_0(5)}$ and $\zeta'_{s_0(5)}$ 123

6.4 Application to the study of the asymptotic behavior of $r_{g_2}(n)$ 124

7 Appendix 127

7.1 SAGE codes 127
# Chapter 1

# Introduction

## 1.1 Notations

**Notation.** Let A, B be sets. We denote the non-strict inclusion as $A \subset B$ (which some authors denote as $A \subseteq B$), and the strict inclusion as $A \subsetneq B$.

**Notation.** Let A be a set, and $B \subset A$ a subset of A, we denote $B^c = A \setminus B$. We denote $|A| \in \mathbb{N}_0 \cup \{+\infty\}$ the cardinal of A.

**Notation.** We use a boldface letter to represent multi-index of the form $\mathbf{x} = (x_a)_{a \in A} \in \mathbb{C}^A$ with A a finite set. If all $x_a$ are integers and if $|A| = n$, we will sometimes call such index a n-tuple integer. Given $B \subset A$, we denote $|\mathbf{x}|_B := \sum_{b \in B} x_b$. If $B = A$, we directly denote $|\mathbf{x}| := |\mathbf{x}|_A$.

**Notation.** Let $1 \le k \le n$ be positive integers. We denote $\mathbf{e}_k = (0, \dots, 0, 1, 0, \dots, 0) \in \mathbb{C}^n$ the vector with a 1 at the k-th component, and 0's elsewhere.

**Notation.** Let A and B be finite sets, and $\mathbf{x} = (\mathbf{x}_a)_{a \in A} \in (\mathbb{N}_0^B)^A$, with $\mathbf{x}_a = (x_{a,b})_{b \in B}$ for all $a \in A$. We define for every $b \in B$,

$$ \mathbf{x}(b) := \sum_{a \in A} x_{a,b}. $$

**Remark 1.1.1.** Let A and B be finite sets, and $\mathbf{x} = (\mathbf{x}_a)_{a \in A} \in (\mathbb{N}_0^B)^A$. The multi-index $\mathbf{x}$ corresponds to a function $\mathbf{x} : A \times B \to \mathbb{N}_0$, $(a, b) \mapsto x_{a,b}$. In particular, if A or B is empty, $\mathbf{x}$ corresponds to the empty function. In this case, for every subset $A \subset A$,

$$ |\mathbf{x}|_A = 0, $$

and for every $b \in B$,

$$ \mathbf{x}(b) = 0. $$

**Notation.** Let $s \in \mathbb{C}$ be a complex number. We write $s = \sigma + i\tau$ where $\sigma$ and $\tau$ are respectively the real and imaginary parts of s.

**Notation.** Let $x \in \mathbb{R}$ be a real number. We denote $H_x$ as the open complex half-plane with abscissa x, and $\bar{H}_x$ as its closure:

$$ H_x := \{s \in \mathbb{C} | \sigma > x\}, $$

$$ \bar{H}_x := \{s \in \mathbb{C} | \sigma \ge x\}. $$

**Notation.** Let $a \in \mathbb{C}$, $r \in \mathbb{R}_+$. We denote $\bar{D}_a(r)$ as the closed disc centered at a with radius r:

$$ \bar{D}_a(r) := \{z \in \mathbb{C} \mid |z - a| \le r\}. $$
**Notation.** Let $s \in \mathbb{C}$ be a complex number, and $k \in \mathbb{N}_0$. We denote $s_{\Delta,k} := (s, \dots, s) \in \mathbb{C}^k$. When there is no ambiguity, we will simply denote $s_{\Delta}$.

**Notation.** Let $n \in \mathbb{N}_0$ be an integer. We call the n-th harmonic number the following rational number

$$h_n := \sum_{k=1}^{n} \frac{1}{k},$$

with the convention $h_0 = 0$.

We will denote $\gamma = 0.5772156649...$ as Euler's constant [OEI23a].

**Definition-Property 1.1.2.** [EMOT81, Chap.1] We denote by $\Gamma$ the Euler's gamma function defined on $H_0$ by the integral

$$\forall s \in H_0, \quad \Gamma(s) = \int_{0}^{+\infty} x^{s-1}e^{-x}dx.$$

The function $\Gamma$ is holomorphic on $H_0$, and has a meromorphic extension to $\mathbb{C}$ with simple poles at the nonpositive integers. We denote this extension also by $\Gamma$. We also denote $\psi$ the digamma function defined on $H_0$ by the relation

$$\forall s \in H_0, \quad \psi(s) = \frac{\Gamma'(s)}{\Gamma(s)}.$$

The function $\psi$ is holomorphic on $H_0$, and has a meromorphic extension to $\mathbb{C}$, with simple poles at the nonpositive integers.

Recall that $\Gamma$ satisfies the following functional equation:

$$\Gamma(s) = (s-1)\Gamma(s-1) \quad (s \notin \mathbb{N}_0),$$

and in the neighborhood of $s = 0$, we have the following asymptotic expansion

$$\Gamma(s) = \frac{1}{s} - \gamma + \frac{1}{2} \left( \gamma^2 + \frac{\pi^2}{6} \right) s + O(s^2).$$

For any nonnegative integers $N$, we have

$$\Gamma(N + 1) = N!.$$

We also recall that the function $\frac{1}{\Gamma}$ is holomorphic on $\mathbb{C}$, and it vanishes at the nonpositive integers.

**Notation.** We define the binomial and multinomial coefficients as follows:

For all $k \in \mathbb{Z}$, and $s \in \mathbb{C}$, we set

$$\binom{s}{k} = \begin{cases} \frac{s(s-1)\dots(s-k+1)}{k!} & \text{if } k \ge 0, \\ 0 & \text{otherwise.} \end{cases}$$

For all integers $\mathbf{k} = (k_1, \dots, k_P)$ such that $n = k_1 + \dots + k_P$, we consider the multinomial coefficient:

$$\binom{n}{\mathbf{k}} = \begin{cases} \frac{n!}{k_1! \dots k_P!} & \text{if } k_p \ge 0 \text{ for all } p \in [1, P], \\ 0 & \text{otherwise.} \end{cases}$$

In this manuscript, we will use the principal determination of the logarithm, and we will write for any complex $z \in \mathbb{C}$, $z = |z|e^{i \arg(z)}$ with $\arg(z) \in [-\pi, \pi]$.

By convention, we will say that a sum over an empty set is zero, and that a product over an empty set is 1.
### 1.2 Some history on zeta and multizeta functions

In this thesis, we will focus on a class of multizeta functions defined by the Dirichlet series

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

with $P \ge 1$, $Q \ge 1$, $\mathbf{d} \in H_0^P$, and $l_q$ being linear forms of rank $P$, dependent on each variable $n_p$ for $1 \le p \le P$. We will study its domain of convergence, its meromorphic extension, and establish formulas for directional values at nonpositive integers of $Z$. We will also give a closed formula for the derivative values at nonpositive integers along the same direction.

In [Kom08], Komori gave a new integral representation formula of the Mordell-Tornheim double zeta function, and derived from this work a formula for the values at nonpositive integers of this zeta function. Further on, he provided an expression for directional values in terms of generalized Bernoulli numbers for a broader class of multizeta functions in [Kom10]. However, these generalized Bernoulli numbers are sometimes difficult to compute. For some generalized multiple zeta functions of Euler-Zagier type, explicit formulas at nonpositive integers along a directions have been computed in [Ono13], and more generally in [EM20], using only classical Bernoulli numbers. The work presented here provides explicit relations in terms of classical Bernoulli polynomials for directional values at nonpositive integers of $Z(\mathbf{s}, \mathbf{s}')$. Additionally, explicit values for the directional derivative at nonpositive integers of $Z(\mathbf{s}, \mathbf{s}')$ are also obtained. These expressions are achieved using a Crandall expansion (c.f. [Cra12, §9.2]). In [BD18], this strategy has already yielded approximations for $\zeta^{AV}(-N)$ and $\partial_s \zeta^{AV}(-N)$, with $\zeta^{AV}$ being the Apostol-Vu type zeta function along the diagonal:

$$\zeta^{AV}(s) = \sum_{n_1, n_2 \ge 1} \frac{1}{n_1^s n_2^s (n_1 + n_2)^s}.$$

In [BD18], Borwein and Tomkins studied a Tornheim-type zeta function:

$$\zeta_P^T(s) := \sum_{n_1, \dots, n_P \ge 1} \frac{1}{n_1^s \dots n_P^s (n_1 + \dots + n_P)^s},$$

and computed the values at nonpositive integers of $\zeta_P^T(s)$, as well as the derivative $\partial_s \zeta_P^T(s)$. In [Ono21], Onodera also studied this class of zeta functions and computed the values of the second derivative with respect to $s$ of $\zeta_P^T(s)$ at nonpositive integers.

From our work, we can derive formulas for the special values of the following two Witten zeta functions, as well as their derivatives:

$$\zeta_{so(5)}(s) := 6^s \sum_{n_1, n_2 = 1}^{+\infty} \frac{1}{n_1^s n_2^s (n_1 + n_2)^s (n_1 + 2n_2)^s} \\ \zeta_{g_2}(s) := 120^s \sum_{n_1, n_2 = 1}^{+\infty} \frac{1}{n_1^s n_2^s (n_1 + n_2)^s (n_1 + 2n_2)^s (n_1 + 3n_2)^s (2n_1 + 3n_2)^s}.$$

Let's recall the Shintani zeta functions

$$\zeta^{Sh}(s, \mathbf{d}|\mathbf{c}_1, \dots, \mathbf{c}_Q) := \sum_{\mathbf{n} \in \mathbb{N}_0^P} \prod_{q=1}^Q \left( d_q + \sum_{p=1}^P c_{q,p} n_p \right)^{-s},$$

with $P, Q \ge 1$ being positive integers, $d_q \in H_0$ and $c_{q,p} \in H_0$ being complex number in $H_0$ for all $1 \le p \le P$ and $1 \le q \le Q$. From our work, we will also get formulas for special values of some Shintani zeta functions, and formulas for the special values of their derivatives as well.
1.2.1 Some results on special values of zeta functions

We aim to provide here a brief summary of results on the special values of the Riemann zeta function, the Hurwitz zeta function, Barnes zeta functions, Euler-Zagier zeta functions, some Witten zeta functions, and Shintani zeta functions. We will also discuss their domains of holomorphy and meromorphy. In particular, we will present results on values at nonpositive integers for some of these functions.

All the zeta functions mentioned in the previous paragraph have a domain of convergence of the form $H_{\sigma_0} = \{s = \sigma + i\tau \in \mathbb{C} | \sigma > \sigma_0\}$ with $\sigma_0 \in \mathbb{R}$, and they are holomorphic on their half-plane of convergence. To give meaning to the value at $s = N \in \mathbb{Z}$ when $N$ is not in the half-plane of convergence for these functions, we must first discuss their meromorphic extension and prove that this extension is regular at the integer we considered.

We first recall the Euler-Maclaurin formula.

**Theorem 1.2.1** (Euler-Maclaurin formula). [Ten15] Let $k \in \mathbb{N}_0$ be a nonnegative integer, and $f: [a, b+1] \to \mathbb{C}$ a function of the class $C^{k+1}$ on $[a, b]$. Then we have that

$$ \sum_{n=a}^{b-1} f(n) = \int_a^b f(t)dt + \sum_{n=0}^{k} \frac{(-1)^{n+1} B_{n+1}}{(n+1)!} (f^{(n)}(b) - f^{(n)}(a)) \\ + \frac{(-1)^k}{(k+1)!} \int_a^b b_{k+1}(x) f^{(k+1)}(x)dx, $$

where $B_n$ is the n-th Bernoulli number, and $b_n(t)$ is the n-th periodised Bernoulli polynomial.

We also recall the expression of the Riemann zeta function.

**Definition 1.2.2.** Let $s \in \mathbb{C}$. The Riemann zeta function is defined as follows

$$ \forall s \in H_1, \quad \zeta(s) = \sum_{n=1}^{+\infty} \frac{1}{n^s}. $$

We will see in the following proposition that this function can be extended over $\mathbb{C}$:

**Proposition 1.2.3.** The Riemann zeta function is holomorphic on $H_1$, and it extends meromorphically over $\mathbb{C}$ with a single pole at $s = 1$, which is simple, with a residue of 1.

The meromorphic extension can be proven using the Euler-Maclaurin formula 1.2.1, and this formula also provides the values of the Riemann zeta function at nonpositive integers in terms of the Bernoulli numbers:

$$ \forall N \in \mathbb{N}_0, \quad \zeta(-N) = (-1)^N \frac{B_{N+1}}{N+1}. $$

In particular, it follows that $\zeta(-2N) = 0$ for all integers $N \in \mathbb{N}$.

Using the functional equation of the Riemann zeta function, we can also find relations between the values of $\zeta$ at positive integers with the values of its derivative at nonpositive integers.

**Theorem 1.2.4.** Let $s \in \mathbb{C} \setminus \{0, 1\}$, we have the functional relation

$$ \zeta(s) = 2^s \pi^{s-1} \sin\left(\frac{\pi s}{2}\right) \Gamma(1-s) \zeta(1-s). $$

In particular, we deduce that, for all $n \in \mathbb{N}$,

$$ \zeta'(-2N) = \frac{(-1)^N (2N)!}{2^{2N+1} \pi^{2N}} \zeta(2N+1). $$

For small values, we also have these results:

$$ \zeta'(0) = -\frac{1}{2} \ln(2\pi), $$

$$ \zeta'(-1) = \frac{1}{12} - \ln(A), $$

where $A = 1.28242712...$ is the Glaisher-Kinkelin constant defined in [OEI23b].
**Definition 1.2.5.** Let $d \in H_0$ be a complex number, the Hurwitz zeta function is defined by

$$ \forall s \in H_0, \quad \zeta(s, d) := \sum_{n=0}^{+\infty} \frac{1}{(n+d)^s}. $$

Note that for $d = 1$, we recover the Riemann zeta function. This zeta function admits a meromorphic extension over $\mathbb{C}$.

**Proposition 1.2.6.** The Hurwitz zeta function is meromorphic on $\mathbb{C}$, with a single pole at $s = 1$, which is a simple pole. Moreover, the residue at that pole is 1.

*Proof.* Let $n \in \mathbb{N}$ be a positive integer. Via Euler-Maclaurin formula, we get that for all $\sigma > 1$,

$$ \zeta(s, d) = \frac{d^{1-s}}{1-s} + \sum_{i=0}^{n} \binom{-s}{i} \frac{(-1)^i B_{i+1}}{i+1} d^{-s-i} \\ + (-1)^n \binom{-s}{n+1} \int_0^{+\infty} b_{n+1}(x)(x+d)^{-s-n-1} dx. \qquad (1.1) $$

The integral in the previous formula converges absolutely for all $\sigma > -n$. Thus, this formula allows for a meromorphic extension of the Hurwitz zeta function for $\sigma > -n$, with a unique pole at $s = 1$, of order 1, and with residue 1. $\square$

We also have results on the values at nonpositive integers of the Hurwitz zeta function (see [Apo76, Theorem 12.13]):

$$ \forall N \in \mathbb{N}_0, \quad \zeta(-N, x) = -\frac{b_{N+1}(x)}{n+1}, $$

where $b_{N+1}(x)$ is the $N+1$-th Bernoulli polynomial. Similarly, it was known by Hurwitz (see [Ber85, Formula (3)]) that

$$ (\partial_s \zeta(s, d))|_{s=0} = \ln(\Gamma(d)) + \zeta'(0) = \ln(\Gamma(d)) - \frac{1}{2} \ln(2\pi) \qquad (1.2) $$

Let us also observe that, for all $s \neq 1$, and for all $d \in H_0$, we have $\zeta(s, d) = d^{-s} + \zeta(s, d + 1)$. Differentiating with respect to $s$, we then find

$$ \partial_s \zeta(s, d)|_{s=-N} = -d^N \ln(d) + \partial_s \zeta(s, d+1)|_{s=-N}. \qquad (1.3) $$

Rewriting the Dirichlet series $\zeta\left(s, \frac{1}{2}\right)$, we can prove that $\zeta\left(s, \frac{1}{2}\right) = (2^s - 1)\zeta(s)$. Therefore, by differentiating both sides of the last equality with respect to $s$, we find that

$$ \partial_s \zeta \left( s, \frac{1}{2} \right) \Big|_{s=-1} = -\frac{\ln(2)}{6} - \frac{1}{2} \zeta'(-1). \qquad (1.4) $$

Miller and Adamchik have established in [MA98] an explicit relation between the values of the derivatives of the Hurwitz zeta function with respect to $s$ at nonpositive integers, with a rational coefficient $d$, and special values of logarithms, polygamma functions, and the Riemann zeta function.

The Hurwitz zeta function admits a functional relationship called Hurwitz's formula. Before stating it, we need to introduce the Lerch zeta function:

**Definition 1.2.7.** Let $s \in \mathbb{C}$ such that $\sigma > 1$, $d \in H_0$, and $z \in \mathbb{C}$ such that $|z| \le 1$. We define

$$ \phi(z, s, d) := \sum_{n \ge 0} \frac{z^n}{(n+d)^s}. $$

This zeta function has a meromorphic extension on $\mathbb{C}$. It is even holomorphic for all $z \neq 1$, and it has a unique, simple pole at $s = 1$ when $z = 1$ [EMOT81, §1.11].
**Theorem 1.2.8** (Hurwitz Formula). [Apo76, §12.7] Let $d \in ]0, 1]$ be a real number, and $s \in \mathbb{C}$ be a complex number such that we have either $d \in ]0, 1[$ and $\text{Re}(s) > 0$, or $d = 1$ and $\text{Re}(s) > 1$. We then get that

$$ \zeta(1-s, d) = \frac{\Gamma(s)}{(2\pi)^s} \left( e^{-i\pi s/2} \phi(e^{2i\pi d}, s, 1) + e^{i\pi s/2} \phi(e^{-2i\pi d}, s, 1) \right). $$

Let's note that the case $d = 1$ in the previous formula corresponds to the functional equation of the Riemann zeta function.

We now mention some results about Barnes zeta functions. We will use some of these results later in the manuscript.

**Definition 1.2.9.** Let $P \in \mathbb{N}_0$ be a nonnegative integer, and $(c_1, \dots, c_P)$ a family of complex numbers in $H_0$, and $d \in H_0$. We call the Barnes zeta function the zeta function of the form

$$ \forall s \in H_P, \quad \zeta^B(s, d|c_1, \dots, c_P) := \sum_{\mathbf{n} \in \mathbb{N}_0^P} \frac{1}{\left(\sum_{p=1}^P c_p n_p + d\right)^s}. $$

Barnes proved that this zeta function has a meromorphic extension over $\mathbb{C}$ with simple poles at some positive integers. More precisely:

**Proposition 1.2.10.** [Bar04] The Barnes zeta function $s \mapsto \zeta^B(s, d|c_1, \dots, c_P)$ has a meromorphic extension over $\mathbb{C}$, with its only singularities being simple poles at $s = 1, \dots, P$.

The derivative value at $s = 0$ of such zeta functions was initially studied by Barnes in the theory of multigamma functions (see [Bar01] and [Bar04]). Later on, Vardi found in [Var88] an explicit formula between the value at $\frac{1}{2}$ of the $n$-th multigamma function and the regularized determinant of the Laplacian of the $n$-sphere. He also gave an explicit formula of the value at $\frac{1}{2}$ of the $n$-th multigamma function using only values at nonpositive integers of the derivative of the Riemann zeta function.

Some of our results will be expressed in terms of derivative values at nonpositive integers of Barnes zeta functions. In particular, we will need a result proved by Sakane and Aoki in [SA22]. We first define

$$ C_{P,x}(t) := (t - x + P - 1)(t - x + P - 2) \dots (t - x + 1) \in (\mathbb{Z}[x])[t] \quad (P \ge 2), $$

and $C_{1,x}(t) = 1$. Let's note that $C_{2,x}(t) = t - x + 1$.

**Theorem 1.2.11.** [SA22, Theorem 4] Let $d \in H_0$, and $\mathbf{c} = (c_1, \dots, c_P) \in \mathbb{Q}^P$ be rational numbers. We write $c_1 = \frac{a_1}{b_1}$, ..., $c_P = \frac{a_P}{b_P}$ with $a_p, b_p \ge 1$ being coprime positive integers for all $1 \le p \le P$. Let $x(\mathbf{c}) := \text{lcm}(a_1, \dots, a_P) / \text{gcd}(b_1, \dots, b_P)$, and $\beta_1 := \frac{x(\mathbf{c})}{c_1}$, ..., $\beta_P := \frac{x(\mathbf{c})}{c_P}$. We then have that

$$ \zeta^B(s, d|c_1, \dots, c_P) = \frac{x(\mathbf{c})^{-s}}{(P-1)!} \sum_{v_1=0}^{\beta_1-1} \dots \sum_{v_P=0}^{\beta_P-1} \sum_{k=0}^{P-1} C_{P, \frac{d+c_1v_1+\dots+c_Pv_P}{x(\mathbf{c})}}^{(k)}(0) \zeta \left( s-k, \frac{d+c_1v_1+\dots+c_Pv_P}{x(\mathbf{c})} \right). $$

In Theorem B, we shall prove a generalized version of the above theorem, using a similar strategy as the one used by Aoki and Sakane in [SA22].
**Example 1.2.12.** We have

$$\zeta^B(s, d|1, 1) = (1 - d)\zeta(s, d) + \zeta(s - 1, d),$$

$$\zeta^B(s, d|1, 2) = 2^{-s} \left[ \left(1 - \frac{d}{2}\right) \zeta\left(s, \frac{d}{2}\right) + \left(1 - \frac{d+1}{2}\right) \zeta\left(s, \frac{d+1}{2}\right) + \zeta\left(s-1, \frac{d}{2}\right) + \zeta\left(s-1, \frac{d+1}{2}\right) \right],$$

$$\zeta^B(s, d|1, 3) = 3^{-s} \left[ \left(1 - \frac{d+2}{3}\right) \zeta\left(s, \frac{d+2}{3}\right) + \left(1 - \frac{d+1}{3}\right) \zeta\left(s, \frac{d+1}{3}\right) + \left(1 - \frac{d}{3}\right) \zeta\left(s, \frac{d}{3}\right) \right. \\ \left. + \zeta\left(s-1, \frac{d+2}{3}\right) + \zeta\left(s-1, \frac{d+1}{3}\right) + \zeta\left(s-1, \frac{d}{3}\right) \right],$$

$$\zeta^B(s, d|2, 3) = 6^{-s} \left[ \left(1 - \frac{d+7}{6}\right) \zeta\left(s, \frac{d+7}{6}\right) + \left(1 - \frac{d+5}{6}\right) \zeta\left(s, \frac{d+5}{6}\right) + \left(1 - \frac{d+4}{6}\right) \zeta\left(s, \frac{d+4}{6}\right) \right. \\ + \left(1 - \frac{d+3}{6}\right) \zeta\left(s, \frac{d+3}{6}\right) + \left(1 - \frac{d+2}{6}\right) \zeta\left(s, \frac{d+2}{6}\right) + \left(1 - \frac{d}{6}\right) \zeta\left(s, \frac{d}{6}\right) \\ \left. \zeta\left(s-1, \frac{d+7}{6}\right) + \zeta\left(s-1, \frac{d+5}{6}\right) + \zeta\left(s-1, \frac{d+4}{6}\right) + \zeta\left(s-1, \frac{d+3}{6}\right) + \zeta\left(s-1, \frac{d+2}{6}\right) \right. \\ \left. + \zeta\left(s-1, \frac{d}{6}\right) \right].$$

We then get the following special values $\zeta^B(0, 2|1, 1) = \frac{5}{12}$, $\zeta^B(0, 3|1, 2) = \frac{11}{24}$, $\zeta^B(0, 4|1, 3) = \frac{19}{36}$, and $\zeta^B(0, 5|2, 3) = \frac{31}{72}$. We also can differentiate the previous formulas in respect to s for $\zeta^B(s, d|c_1, c_2)$. We then get

$$(\zeta^B)'(0, 2|1, 1) = -\zeta'(0, 2) + \zeta'(-1, 2),$$

$$(\zeta^B)'(0, 3|1, 2) = -\frac{11 \ln(2)}{24} - \frac{1}{2} \zeta'\left(0, \frac{3}{2}\right) - \zeta'(0, 2) + \zeta'\left(-1, \frac{3}{2}\right) + \zeta'(-1, 2),$$

$$(\zeta^B)'(0, 4|1, 3) = -\frac{19 \ln(3)}{36} - \zeta'(0, 2) - \frac{2}{3} \zeta'\left(0, \frac{5}{3}\right) - \frac{1}{3} \zeta'\left(0, \frac{4}{3}\right) + \zeta'(-1, 2) + \zeta'\left(-1, \frac{5}{3}\right) + \zeta'\left(-1, \frac{4}{3}\right),$$

$$(\zeta^B)'(0, 5|2, 3) = -\frac{31 \ln(6)}{72} - \zeta'(0, 2) - \frac{2}{3} \zeta'\left(0, \frac{5}{3}\right) - \frac{1}{2} \zeta'\left(0, \frac{3}{2}\right) - \frac{1}{3} \zeta'\left(0, \frac{4}{3}\right) - \frac{1}{6} \zeta'\left(0, \frac{7}{6}\right) + \frac{1}{6} \zeta'\left(0, \frac{5}{6}\right) \\ + \zeta'(-1, 2) + \zeta'\left(-1, \frac{5}{3}\right) + \zeta'\left(-1, \frac{3}{2}\right) + \zeta'\left(-1, \frac{4}{3}\right) + \zeta'\left(-1, \frac{7}{6}\right) + \zeta'\left(-1, \frac{5}{6}\right),$$

where we noted $(\zeta^B)' = \partial_s \zeta^B$, and $\zeta'(s, d) = \partial_s \zeta(s, d)$.

Shintani introduced zeta functions that allow him to study Dedekind zeta functions associated with a totally real number field (see [Shi76b], [Shi76a], [Shi77a], [Shi77b], [Shi77c], [Shi80]). These functions, in particular, generalize Barnes' zeta functions:

**Definition 1.2.13.** Let $P, Q \in \mathbb{N}_0$ be nonnegative integers, $c = (c_{q,p})_{1 \le p \le P, 1 \le q \le Q}$ and $(d_q)_{1 \le q \le Q}$ be two families of complex numbers in $H_0$. We call the following function the Shintani zeta function:

$$\zeta^{Sh}(s, \mathbf{d}|\mathbf{c}) = \sum_{n_1, \dots, n_P \ge 0} \prod_{q=1}^{Q} (c_{q,1} n_1 + \dots + c_{q,P} n_P + d_q)^{-s}.$$

Note that, for $Q = 1$, we recover the definition of Barnes' zeta function. This zeta function is holomorphic on $H_{P/Q}$ and has a meromorphic extension over $\mathbb{C}$. Shintani proved that this function has the same poles as those of the function $\frac{\Gamma(Ps - Q)}{\Gamma(s)}$. He also proved certain formulas at the nonpositive integers of these functions. A part of our work particularly allows us to study the values of some of these functions at nonpositive integers, as well as the values of their derivatives.
We deduce from the above that the zeta functions of Barnes and Shintani are regular at nonpositive integers. The values of their derivatives with respect to the variable s at s = 0 allow us to define multigamma functions, introduced by Barnes in [Bar04], and then generalized by Friedmann and Ruijsenaars.

**Definition 1.2.14.** Let $P \in \mathbb{N}_0$ be a nonnegative integer, and $(c_1, ..., c_P)$ be complex numbers in $H_0$, and $d \in H_0$. We call Barnes multigamma function the following

$$ \Gamma_P(d|c_1, ..., c_P) := \exp((\partial_s \zeta^B(s, d|c_1, ..., c_P)))|_{s=0}). $$

**Example 1.2.15.** Let $d \in H_0$, we have that

$$ \Gamma_0(d) = \frac{1}{d} \\ \Gamma_1(d|1) = \frac{1}{\sqrt{2\pi}} \Gamma(d), $$

where $\Gamma$ is Euler's gamma function.

Barnes also obtains the following functional relation

$$ \Gamma_P(d|c_1, ..., c_P) = \Gamma_{P-1}(d|c_1, ..., c_{P-1})\Gamma_P(d+c_P|c_1, ..., c_P), $$

which generalizes the classical recurrence relation of Euler's gamma function.

Through Shintani-type zeta functions, Friedman and Ruijsenaars introduced in [FR04] Shintani-type multigamma functions, using a construction similar to the one used by Barnes.

**Definition 1.2.16.** Let $P, Q \in \mathbb{N}_0$ be nonnegative integers, $\mathbf{c} = (c_{q,p})_{1 \le p \le P, 1 \le q \le Q}$ and $(d_q)_{1 \le q \le Q}$ be two families of complex numbers in $H_0$. The Shintani multigamma function is defined as:

$$ \Gamma_P(\mathbf{d}|\mathbf{c}_1, ..., \mathbf{c}_P) := \exp((\partial_s \zeta^{Sh}(\mathbf{s}, \mathbf{d}|\mathbf{c}_1, ..., \mathbf{c}_q)))|_{s=0}), $$

where $\mathbf{c}_q = (c_{q,p})_{p \in \{1, P\}}$.

In [FR04], Friedman and Ruijsenaars studied these Shintani multigamma functions to deduce Raabe-type formulas, generalizing the following Raabe formula:

$$ \int_{0}^{1} \ln(\Gamma(x))dx = \ln(\sqrt{2\pi}). $$

We will now discuss a class of zeta functions related to Lie algebras. Witten introduced in [Wit91] the series

$$ \sum_{\rho} \frac{1}{\dim(\rho)^{2g}}, $$

where $g$ is a positive integer, and $\rho$ runs over the irreducible finite representations, up to isomorphism, of a Lie algebra $\mathfrak{g}$, and $g \in \mathbb{N}$. Witten particularly proved that

$$ \sum_{\rho} \frac{1}{\dim(\rho)^{2g}} \in \pi^{2g}\mathbb{Q}. $$

Zagier then introduced the following Witten zeta function :

**Definition 1.2.17.** [Zag94] Let $\mathfrak{g}$ be a semi-simple Lie algebra. We define a Witten zeta function attached to $\mathfrak{g}$ the following zeta function:

$$ \zeta_{\mathfrak{g}}(s) := \sum_{\rho} \dim(\rho)^{-s}, $$

where $\rho$ runs over the finite irreducible representations, up to isomorphism, of the Lie algebra $\mathfrak{g}$.
We can introduce $r_g(n)$ as the number of isomorphism classes among the representations of dimension $n$.

**Definition 1.2.18.** Let $\mathfrak{g}$ be a semi-simple Lie algebra. We set $r_{\mathfrak{g}}(n)$ as the number of representations of $\mathfrak{g}$ of dimension $n$, up to isomorphism.

It is well-known that a representation of a semi-simple Lie algebra uniquely decomposes into a direct sum of irreducible representations. Thus, for a representation $\rho$ of $\mathfrak{g}$, we have that there exist integers $k_1, \dots, k_m$ and irreducible representations $\phi_1, \dots, \phi_m$ such that

$$ \dim(\rho) = \sum_{i=1}^{m} k_i \dim(\phi_i). $$

More specifically, for the Lie algebras $\mathfrak{so}(5)$ and $\mathfrak{g}_2$, we have:

**Example 1.2.19.** [Hum72, §24.3]

1) The set of isomorphism class of irreducible representations of $\mathfrak{sl}(3)$ is of the form $(\phi_{i,j})_{i,j \ge 1}$, with

$$ \dim(\phi_{i,j}) = \frac{ij(i+j)}{2}. $$

Then we get the following expression for the number of irreducible representations of dimension $n$ of $\mathfrak{sl}(3)$:

$$ \left| \left\{ (i, j) \in \mathbb{N}^2 \middle| \frac{ij(i+j)}{2} = n \right\} \right|, $$

and we get the following expression of the number of representations $r_{\mathfrak{sl}(3)}(n)$ of $\mathfrak{sl}(3)$ is

$$ \begin{aligned} r_{\mathfrak{sl}(3)}(n) &= \left| \left\{ (k_{i,j})_{i,j \ge 1} \in (\mathbb{N}_0)^{\mathbb{N}^2} \middle| \sum_{i,j \ge 1} k_{i,j} \dim(\phi_{i,j}) = n \right\} \right|, \\ &= \left| \left\{ (k_{i,j})_{i,j \ge 1} \in (\mathbb{N}_0)^{\mathbb{N}^2} \middle| \sum_{i,j \ge 1} k_{i,j} \frac{ij(i+j)}{2} = n \right\} \right|. \end{aligned} $$

2) The set of isomorphism class of irreducible representations of $\mathfrak{so}(5)$ is of the form $(\phi_{i,j})_{i,j \ge 1}$, with

$$ \dim(\phi_{i,j}) = \frac{ij(i+j)(i+2j)}{3!}. $$

Then we get the following expression for the number of irreducible representations of dimension $n$ of $\mathfrak{so}(5)$:

$$ \left| \left\{ (i, j) \in \mathbb{N}^2 \middle| \frac{ij(i + j)(i + 2j)}{3!} = n \right\} \right|, $$

and we get the following expression of the number of representation $r_{\mathfrak{so}(5)}(n)$ of $\mathfrak{so}(5)$:

$$ \begin{aligned} r_{\mathfrak{so}(5)}(n) &= \left| \left\{ (k_{i,j})_{i,j \ge 1} \in (\mathbb{N}_0)^{\mathbb{N}^2} \middle| \sum_{i,j \ge 1} k_{i,j} \dim(\phi_{i,j}) = n \right\} \right|, \\ &= \left| \left\{ (k_{i,j})_{i,j \ge 1} \in (\mathbb{N}_0)^{\mathbb{N}^2} \middle| \sum_{i,j \ge 1} k_{i,j} \frac{ij(i+j)(i+2j)}{3!} = n \right\} \right|. \end{aligned} $$
3) The set of isomorphism class of irreducible representations of g_2 is of the form $(\phi_{i,j})_{i,j \ge 1}$, with

$$ \dim(\phi_{i,j}) = \frac{ij(i+j)(i+2j)(i+3j)(2i+3j)}{5!}. $$

Then we get the following expression for the number of irreducible representations of dimension n of g_2:

$$ \left| \left\{ (i,j) \in \mathbb{N}^2 \middle| \frac{ij(i+j)(i+2j)(i+3j)(2i+3j)}{5!} = n \right\} \right|, $$

and we get the following expression of the number of representation $r_{g_2}(n)$ of g_2:

$$ \begin{aligned} r_{g_2}(n) &= \left| \left\{ (k_{i,j})_{i,j \ge 1} \in (\mathbb{N}_0)^{\mathbb{N}^2} \middle| \sum_{i,j \ge 1} k_{i,j} \dim(\phi_{i,j}) = n \right\} \right| \\ &= \left| \left\{ (k_{i,j})_{i,j \ge 1} \in (\mathbb{N}_0)^{\mathbb{N}^2} \middle| \sum_{i,j \ge 1} k_{i,j} \frac{ij(i+j)(i+2j)(i+3j)(2i+3j)}{5!} = n \right\} \right|. \end{aligned} $$

Let's write $f_g(n)$ the number of irreducible representations of dimension $n$ of a semi-simple Lie algebra g. Then by definition of the Witten zeta function, we get that

$$ \zeta_g(s) = \sum_{n=1}^{+\infty} \frac{f_g(n)}{n^s}. $$

We will see in Chapter 6 that this zeta function can be simplified using a product of linear forms, thanks to Weyl's Theorem 6.2.9.

**Example 1.2.20.** [Zag94], [KMT24] Let $sl_{n+1} = \{X \in M_{n+1}(\mathbb{C}) | \text{tr}(X) = 0\}$, and $so_{2n+1} = \{X \in M_{2n+1}(\mathbb{C}) | {}^t X + X = 0\}$, where Tr denotes the trace, and {}$^t X$ denotes the transpose of a matrix. For these Lie algebras, we can associate Witten's zeta functions:

$$ \begin{aligned} \zeta_{sl(2)}(s) &= \zeta(s), \\ \zeta_{sl(3)}(s) &= 2^s \sum_{n_1, n_2 \ge 1} \frac{1}{n_1^s n_2^s (n_1 + n_2)^s}, \\ \zeta_{so(5)}(s) &= 6^s \sum_{n_1, n_2 \ge 1} \frac{1}{n_1^s n_2^s (n_1 + n_2)^s (n_1 + 2n_2)^s}. \end{aligned} $$

For the exceptional Lie algebra g_2 described in [Bou81], we get that

$$ \zeta_{g_2}(s) = 120^s \sum_{n_1, n_2 \ge 1} \frac{1}{n_1^s n_2^s (n_1 + n_2)^s (n_1 + 2n_2)^s (n_1 + 3n_2)^s (2n_1 + 3n_2)^s}. $$

Note that in the literature, we sometimes find multivariable variants of these zeta functions in the works of Matsumoto, Komori, Yasushi, and Tsumura [KMT10c], [KMT10b], [KMT12], [KMT11b]. These authors provide an expression for the set of singularities of these Witten multizeta functions. This allows us in particular to obtain candidate poles for the corresponding univariable Witten zeta function.

* From [KMT10d, Theorem 6.2], we deduce that the zeta function $\zeta_{so(5)}(s)$ has poles of the form

$$ s = \frac{1}{2}, \quad s = \frac{1-k}{3} \quad (k \in \mathbb{N}_0). \qquad (1.5) $$

* From [KMT11a, Theorem 3.1], we deduce that the zeta function $\zeta_{g_2}(s)$ has poles of the form

$$ s = \frac{1}{3}, \quad s = \frac{1-k}{5} \quad (k \in \mathbb{N}_0). \qquad (1.6) $$
We will show via Proposition 2.2.4 that nonpositive integers are not poles for $\zeta_{s_0(5)}(s)$ and for $\zeta_{g_2}(s)$.

These zeta functions are well-suited for studying the representations of semi-simple Lie algebras, and one can express their special values to obtain information about these representations. Indeed, recent work by Romik [Rom17] allows us to study the asymptotic behavior of the number representations of dimension $n$ of $sl(3)$ in terms of the values $\zeta_{s_1(3)}(0)$ and $\zeta'_{s_1(3)}(0)$, as well as the residues of $\zeta_{s_1(3)}(s)$ at the singularities of $\zeta_{s_1(3)}$. Kurokawa and Ochiai in [KO13] have studied the values at nonpositive integers of $\zeta_{s_1(3)}(s)$, and have deduced results for the values of certain $p$-adic Witten zeta functions.

Onodera subsequently obtained other formulas by studying a Mordell-Tornheim-type zeta function:

$$ \zeta_P^{MT}(s) := \sum_{\mathbf{n} \in \mathbb{N}_0^P} \frac{1}{(n_1 + d_1)^s \dots (n_P + d_P)^s (n_1 + d_1 + \dots + n_P + d_P)^s}, $$

with $P \ge 2$ an integer. Note that by setting $P = 2$, we obtain $2^{-s}\zeta_{s_1(3)}(s)$.

Onodera computed the values of this zeta function at nonpositive integers via a Crandall expansion. This strategy consists of partitioning the integration domain of an integral representation of the zeta function whose values one is looking to study, with the domain depending on a free variable. More precisely, Onodera obtained the following expression:

$$ \zeta_P^{MT}(s, \mathbf{d}) = \frac{1}{\Gamma(s)} \sum_{\mathbf{n} \in \mathbb{N}_0^P} \frac{1}{(n_1 + d_1)^s \dots (n_P + d_P)^s (n_1 + d_1 + \dots + n_P + d_P)} \int_{\theta(n_1+d_1+\dots+n_P+d_P)}^{+\infty} e^{-y} y^{s-1} dy \\ + \frac{1}{\Gamma(s)} \int_0^\theta \prod_{p=1}^P e^{-d_p y_p} \phi(e^{-y}, s, d_p) dy_1 \dots dy_P, $$

with $\phi$ being the Lerch zeta function described in Definition 1.2.7, and where $\theta$ is a sufficiently small real free variable. The first term on the right-hand side of the previous equation is holomorphic and vanishes at nonpositive integers. The second term can be re-expressed via the Erdélyi formula 2.6, and Onodera ultimately obtains the following formulas:

**Theorem 1.2.21.** [Ono21, Theorem 3, Theorem 4] Let $\mathbf{d} \in \mathbb{R}_*^{+P}$ and $t \in [1, P-1]$ such that $t < |\mathbf{d}|$. For all nonnegative integers $N \in \mathbb{N}_0$, we have that

$$ \zeta_P^{MT}(-N, \mathbf{d}) = \sum_{P \subseteq [1, P]} \frac{(-1)^{(N+1)(|\mathcal{P}|)} N!^{|\mathcal{P}|+1}}{|\mathcal{P}|+1} \sum_{\substack{\mathbf{k}=(k_p)_{p \in \mathcal{P}} \in \mathbb{N}_0^{\mathcal{P}} \\ |\mathbf{k}|_{|\mathcal{P}|}=(|\mathcal{P}|+1)N+|\mathcal{P}|}} \prod_{p \in \mathcal{P}} \frac{\zeta(-N-k_p, d_p)}{k_p!} $$

$$ \zeta_P^{MT'}(-N, \mathbf{d}) = \sum_{P \subseteq [1, P]} (-1)^{|\mathcal{P}|(|N+1|-1)N!^{|\mathcal{P}|}} \left( \sum_{\substack{\mathbf{k}=(k_p)_{p \in P} \in \mathbb{N}_0^P \\ u \ge 0 \\ |\mathbf{k}|_{|P|+u}=(|\mathcal{P}^c|+1)N+|\mathcal{P}|}} \prod_{p \in \mathcal{P}^c} \frac{\zeta(-N-k_p, d_p)}{k_p!} \right) \cdot \left( \frac{(-1)^N \zeta'(-N-u, |\mathbf{d}|-t)}{u!} + \frac{1}{|\mathcal{P}^c|} \sum_{p \in \mathcal{P}^c} \frac{(-1)^N \zeta'(-N-u, d_p)}{u!} \right), $$

where we noted $\zeta_P^{MT'}(s, \mathbf{d}) := \partial_s \zeta_P^{MT}(s, \mathbf{d})$.

Onodera also obtains values for the second derivatives [Ono21, Theorem 5], which we will not detail here. In this thesis, we will aim to generalize the above theorem for a broader class of zeta functions. This will specifically allow us to obtain values at nonpositive integers for $\zeta_{s_0(5)}(s)$ and $\zeta_{g_2}(s)$, as well as those for their first derivatives.
1.2.2 Some results on special values of multizeta functions

We will provide some results here on the special values of certain multizeta functions related to products of linear forms. In particular, we will give expressions of these values for Euler-Zagier multizeta functions:

**Definition 1.2.22.** Let $P \in \mathbb{N}$, and $\mathbf{c} = (c_1, \dots, c_P) \in H_0^P$, $\mathbf{d} = (d_1, \dots, d_P) \in \mathbb{C}^P$ be complex numbers such that $\mathrm{Re}(d_i) > -\mathrm{Re}(c_1)$ for all $p \in [1, P]$. The generalized Euler-Zagier multizeta function is defined as

$$ \zeta_P^{EZ}(\mathbf{s}, \mathbf{d}) = \sum_{\substack{n_1 \ge 1 \\ n_2, \dots, n_P \ge 0}} \prod_{p=1}^{P} (c_1 n_1 + \dots + c_P n_P + d_p)^{-s_p}. $$

We denote $\zeta_P^{EZ}(\mathbf{s}) := \zeta(\mathbf{s}, \mathbf{1}, (0, 1, \dots, P-1))$ the Euler-Zagier multizeta function.

Note that, through a change of variables, we can rewrite the "classical" Euler-Zagier zeta function in the following form,

$$ \zeta_P^{EZ}(\mathbf{s}) = \sum_{0 < n_1 < \dots < n_P} \frac{1}{n_1^{s_1} \dots n_P^{s_P}}. $$

**Proposition 1.2.23.** The generalized Euler-Zagier multizeta function $\zeta_P^{EZ}(\mathbf{s}, \gamma, \mathbf{d})$ is holomorphic on $\{(s_1, \dots, s_P) \in \mathbb{C}^P | \sigma_k + \dots + \sigma_P > P + 1 - k, 1 \le k \le P\}$, and is meromorphic on $\mathbb{C}^P$, with poles included in hyperplanes of the form

$$ s_p + \dots + s_P = (P + 1 - p) - k_p, \quad (1 \le p \le P, k_1, \dots, k_P \in \mathbb{N}_0). $$

From the previous proposition, we see that the quantity $\zeta_P^{EZ}(-\mathbf{N}, \mathbf{d})$ does not always make sense. Moreover, a result from [AET01] specifies that most of the nonpositive integers are actually singularities for $\zeta_P^{EZ}(\mathbf{s})$. To still study the values of this function, Akiyama, Egami, and Tanigawa investigated the "regular" values of this zeta function by defining

$$ \zeta_P^{EZ,reg}(-\mathbf{N}) := \lim_{s_1 \to -N_1} \dots \lim_{s_P \to -N_P} \zeta_P^{EZ}(\mathbf{s}). $$

The authors also introduced "non-regular" values by reversing the previous limits, defining

$$ \zeta_P^{EZ,non-reg}(-\mathbf{N}) := \lim_{s_P \to -N_P} \dots \lim_{s_1 \to -N_1} \zeta_P^{EZ}(\mathbf{s}). $$

In [AET01] and in [AT01], the authors obtained a recursive formula between the regular values $\zeta_P^{EZ,reg}(-\mathbf{N})$ (respectively $\zeta_P^{EZ,non-reg}(-\mathbf{N})$ with $\mathbf{N} \in \mathbb{N}_0^P$, and the regular values of the form $\zeta_{P-1}^{EZ,reg}(-\mathbf{N}')$ (respectively $\zeta_{P-1}^{EZ,non-reg}(-\mathbf{N}')$, with $\mathbf{N}' \in \mathbb{N}_0^{P-1}$ to obtain an explicit expression of regular and non-regular values at nonpositive integers. These results were later generalized by Sasaki in [Sas09a] and in [Sas09b] to different orders of limits than those of regular and non-regular values at nonpositive tuples.

We can also study the "directional" values of this multizeta function. In [Kom10], Komori introduces the notion of directional value for such multizeta functions and provides an expression for these in the case of Hurwitz-Lerch multizeta functions. Komori then deduces formulas for the values at nonpositive integers of the Shintani and Euler-Zagier zeta functions, using generalized Bernoulli numbers (often not explicit).

**Definition 1.2.24.** Let $P, Q \in \mathbb{N}$ be integers, $\mathbf{d} = (d_1, \dots, d_Q)$, $\mathbf{c} = (c_{q,p})_{(q,p) \in [1,Q] \times [1,P]}$ and $\xi = (\xi_1, \dots, \xi_P)$ be complex numbers such that $c_{q,p} \in \bar{H}_0$, $d_q \in H_0$ and $\xi_q \in \mathbb{T} = \mathbb{C}/2i\pi\mathbb{Z}$. We call the function $\zeta^{HL}$ the Hurwitz-Lerch multizeta function, defined as

$$ \zeta^{HL}(\xi, \mathbf{d}, \mathbf{c}, \mathbf{s}) = \sum_{n_1=0}^{+\infty} \dots \sum_{n_P=0}^{+\infty} \frac{e^{\xi_1 n_1} \dots e^{\xi_P n_P}}{(c_{1,1} n_1 + \dots + c_{1,P} n_P + d_1)^{s_1} \dots (c_{Q,1} n_1 + \dots + c_{Q,P} n_P + d_Q)^{s_Q}}. $$

We also define the directional values of $\zeta^{HL}$, following a direction $\mu \in \mathbb{C}^P$:

$$ \zeta^{HL}(\xi, \mathbf{d}, \mathbf{c}, -\mathbf{N}_{\mu}) := \lim_{s \to 0} \zeta^{HL}(\xi, \mathbf{d}, \mathbf{c}, -\mathbf{N} + s\mu). $$
Komori shows that this multizeta function is holomorphic on $\{s \in \mathbb{C} | \forall q \in [1, Q], \text{Re}(s_q) > \sigma_{0,q}\}$, where $\sigma_{0,q} > 0$ for all $q \in [1, Q]$. He also proves that this multizeta function has a meromorphic extension on $\mathbb{C}^Q$, and that the previously defined directional values make sense. Before briefly summarizing his results, we must assume the following conditions on the data of the multizeta function $\zeta^{HL}$.

Let's suppose that there exist an integer $P_0 \in [1, P]$ and a set $\mathcal{B} \subset [1, Q] \times [1, P_0]$ such that

$$ \begin{cases} c_{q,p} = 0 & \text{if and only if } (q, p) \in \mathcal{B}^c \\ \xi_p = 0 & \text{if and only if } p \in [1, P_0], \end{cases} $$

with $\mathcal{B}^c = [1, Q] \times [1, P_0] \setminus \mathcal{B}$. Let $\mathcal{B}_{\bullet p} := \{q | (q, p) \in \mathcal{B}\}$. Let's suppose that $\mathcal{B}_{\bullet p} \neq \emptyset$ for all $p \in [1, P_0]$.

Komori gave in [Kom10] a surface integral representation (i.e an integral representation on a surface) of the multizeta function $\zeta^{HL}(\xi, \mathbf{d}, \mathbf{c}, \mathbf{s})$. This surface integral representation particularly generalizes the usual integral representation of Riemann's zeta function

$$ \zeta(s) = \frac{1}{\Gamma(s)(e^{2i\pi s} - 1)} \int_C \frac{z^{s-1}}{e^z - 1} dz, $$

where $C$ denotes the Hankel's contour [EMOT81, §1.10]. It is good to note that this integral representation of $\zeta(s)$ particularly allows for obtaining a formula for the values of the Riemann zeta function at nonpositive integers using Bernoulli numbers, by using

$$ \frac{z}{e^z - 1} = \sum_{k=0}^{+\infty} B_k \frac{z^k}{k!}. $$

In the same vein, Komori subsequently used the surface integral representation obtained in [Kom10, Theorem 3.14] to get an expression for the directional values in terms of generalized Bernoulli numbers $B_w(\xi, \mathbf{d}, \mathbf{c}, \mathbf{k})$ with $\mathbf{k} \in \mathbb{N}_0^Q$. To see the construction of these numbers, we refer to [Kom10, Definition 3.20].

**Theorem 1.2.25.** [Kom10, Theorem 3.14, Corollary 3.16] Under the above assumptions, the multizeta function $\zeta^{HL}(\xi, \mathbf{d}, \mathbf{c}, \mathbf{s})$ is meromorphic in $\mathbf{s}$ on all $\mathbb{C}^P$. Moreover, his singularities lie in hyperplanes of the form

$$ \left\{ \begin{array}{ll} \sum_{q \in \mathcal{Q}} s_q = Q - k & (\mathcal{Q} \subset [1, Q], |\mathcal{Q}| \ge 2, k \in \mathbb{N}_0) \\ s_q = 1, \dots, P & (1 \le q \le Q). \end{array} \right. $$

Moreover, Komori shows in [Kom10, Theorem 3.22] that the directional values $\zeta^{HL}(\xi, \mathbf{d}, \mathbf{c}, -\mathbf{N}_\mu)$ are well-defined, and he provides a non-explicit expression for them, imposing a non-vanishing condition on the sum of the directions. Incidentally, it turns out that if one manages to re-express the generalized Bernoulli numbers in the formula obtained by Komori, then this formula can be used to determine whether a nonpositive tuple is a singularity for $\zeta^{HL}$. Indeed, if one obtains two distinct values of $\zeta^{HL}$ at a nonpositive tuple, along two different directions, this particularly implies that the nonpositive tuple is a singularity.

**Remark 1.2.26.** The directional formula [Kom10, Theorem 3.22] obtained by Komori is not very explicit, as it is sometimes difficult to get at a satisfactory expression for the generalized Bernoulli numbers he introduces in his article.

In [EM20], Essouabri and Matsumoto studied the directional values of the generalized Euler-Zagier multizeta function. Here, we will only provide a very simplified and much less detailed formula:

**Theorem 1.2.27.** [EM20, Theorem 1] Let $\mathbf{N} \in \mathbb{N}_0^P$, and $\mathbf{c} = (c_1, \dots, c_P) \in \mathbb{C}^P$ such that $\text{Re}(\mu_p + \dots + \mu_P) \neq 0$ for all $p \in [1, P]$. Then $\zeta_P^{EZ}(-\mathbf{N}_\mu, \mathbf{c}, \mathbf{d}) := \lim_{s \to 0} \zeta_P^{EZ}(-\mathbf{N} + s\mu, \mathbf{c}, \mathbf{d})$ exists, and we have that:

$$ \zeta_P^{EZ}(-\mathbf{N}_\mu, \mathbf{c}, \mathbf{d}) = \sum_{\mathbf{k}=(k_1, \dots, k_P) \in [0, |\mathbf{N}|+P+1]^P} P_{\mathbf{k}}(\mathbf{b}, \mathbf{c}, \mu) \prod_{i=1}^{P} B_{k_i}, $$
with $B_n$ being the n-th Bernoulli number, and $P_k(\mathbf{b}, \mathbf{c}, \mu)$ an explicit polynomial in $c_p, d_p$ and $\mu_p$ for $p \in [1, P]$.

**Remark 1.2.28.** 1) In [EM21], Essouabri and Matsumoto generalized this last result for a larger class of multizeta functions, which are of the form

$$ \zeta_m(\mathbf{s}, \mathbf{P}) = \sum_{n_1, \dots, n_m} \frac{1}{\prod_{j=1}^m P_j(n_1, \dots, n_j)^{s_j}}, $$

where $P_j$ is a polynomial in $X_1, \dots, X_j$ for all $1 \le j \le m$, such that they respect the H0S condition (see [Ess97]), where $P_m$ is homogeneous and elliptic and such that

$$ \forall 1 \le j \le m, \quad \lim_{x_1 + \dots + x_j \to +\infty} P_j(x_1, \dots, x_j) = +\infty. $$

2) In [MW02], Matsumoto and Weng studied zeta functions of the form

$$ \zeta^{\text{MW}}(s) = \sum_{n=1}^{+\infty} \frac{P(n)}{Q(n)^s}, $$

with $P$ and $Q$ two nonzero polynomials such that $Q$ doesn't vanish at positive integers. They provided an explicit formula for its derivative value at $s = 0$. This formula is of particular interest when studying the regularized determinant of the Laplacian of the n-sphere (see [Var88], [WY96], [QC96], [Kum99]).

### 1.2.3 On asymptotic formulas for some partitions

We will explain in this subsection some arithmetic applications of computing special values of Witten's zeta functions, and their derivatives. We will also discuss Meinardus' theorem on the asymptotic behavior of some partition numbers generated by infinite product. We will also discuss some recent work from Bridges, Brindle, Bringmann, Craig, Franke, Males on asymptotic formulas for more general partition numbers in the preprints [BB23], [BBF23], [BCM23]. We will give in particular the statements of the results obtained by Bridges, Bringmann, Brindle, Franke in [BBBF23], and we will talk about their approach.

In combinatorics, it is standard to use a generating series of the form

$$ \omega(q) = \sum_{n \ge 0} r(n)q^n $$

in order to study the behavior of a sequence of real numbers $r(n)$. If the function $\omega$ is holomorphic for $|q| < 1$, then by Cauchy's theorem we have

$$ r(n) = \frac{1}{2i\pi} \int_C \frac{\omega(q)}{q^{n+1}} dq, $$

where $C$ is a circle centered at zero running counterclockwise, and included in the unit disk. In the general case, no general result exists to obtain an asymptotic formula for a sequence $r(n)$. In the rest of this section, we will focus on sequences obtained by considering a generating series written as an infinite product.

We consider a function $G_f(q)$ of the form

$$ G_f(q) = \prod_{n=1}^{+\infty} \frac{1}{(1 - q^n)^{f(n)}}, $$

where $f(n)$ are real numbers. By distributing the product, we obtain

$$ G_f(q) = \sum_{n=0}^{+\infty} p_f(n)q^n. \qquad (1.7) $$
We consider the Dirichlet series

$$L_f(s) = \sum_{n=1}^{+\infty} \frac{f(n)}{n^s} \quad (s = \sigma + i\tau, \sigma > \sigma_0),$$

with $\sigma_0$ its abscissa of convergence.

**Example 1.2.29.** [And76, 5.1.2] The simplest infinite product one can think of corresponds to studying the number of partitions $p(n)$ of a positive integer $n$:

$$\prod_{n=1}^{+\infty} \frac{1}{1-q^n} = \sum_{n=1}^{+\infty} p(n)q^n.$$

The corresponding Dirichlet series is Riemann's zeta function. Hardy and Ramanujan showed that as $n \to +\infty$,

$$p(n) \sim \frac{1}{4n\sqrt{3}} \exp\left(\pi\sqrt{\frac{2n}{3}}\right).$$

We now set

$$g_f(z) := \sum_{n=1}^{+\infty} f(n)e^{-nz}.$$

We assume the following conditions:

i) $\sigma_0 > 0$.

ii) $L_f$ is meromorphic continuation to $H_{-C_0}$, with $0 < C_0 < 1$ and such that $L_f$ has only one pole at $s = \sigma_0$ in this domain, and that this pole is of order 1 and has residue $A$.

iii) We have the following upper bound as $\tau \to +\infty$, and uniform in $\sigma > -C_0$,

$$|L_f(s)| = O(|\tau|^{C_1}),$$

with $C_1 > 0$ and $s = \sigma + i\tau$.

iv) Let $\epsilon > 0$. For all $z = x + 2i\pi y$ such that $\arg(z) > \frac{\pi}{4}$, $|x| \le \frac{1}{2}$ and $|y| \ll 1$, we have

$$\mathrm{Re}(g_f(z)) - g_f(x) \le -C_2 x^{-\epsilon},$$

with $C_2 > 0$ a constant depending on $\epsilon$.

Meinardus' theorem then provides an asymptotic relation for the sequence $p_f(n)$.

**Theorem 1.2.30 (of Meinardus).** [And76, Theorem 6.2] As $n \to +\infty$, we have

$$p_f(n) = C n^{\kappa} \exp \left( n^{\frac{\alpha}{\alpha+1}} \left(1 + \frac{1}{\alpha}\right) \left(A\Gamma(\alpha+1)\zeta(\alpha+1)\right)^{\frac{1}{\alpha}} \right) \left(1 + O(n^{-\kappa_1})\right),$$

where $p_f(n)$ are defined in formula 1.7, and $C$, $\kappa$, $\kappa_1$ are real numbers depending on $L_f(0)$, $L'_f(0)$, $C_0$, and $\alpha$.

The proof of this theorem relies on the saddle point method, key aspects of which will be explained here. For details of the method, we refer to [FS09, VIII.3]. The method takes advantage of the fact that the integral $\frac{1}{2i\pi} \int_C \frac{G_f(q)}{q^{n+1}} dq$ is independent of the radius of circle $C$ within the unit disk. This independence can be exploited by judiciously choosing an appropriate radius $\rho_0(n)$. The saddle point
method involves selecting this radius in such a way as to minimize $\left|\frac{G_f(q)}{q^n}\right|$ over a minor arc $C_1(\theta_0(n))$.

Specifically, we choose such a saddle point so that the derivative of the integrand at $q = \rho_0(n)$ is $O(n^\delta)$, where $\delta$ is well-chosen. Alternatively, we can find $\rho_n$ by studying the derivative of the function

$$h_f(z) := \ln(L_f(z)) - n \log(z).$$

We then look for $\rho_n$ such that $h'_f(\rho_0(n)) = -n + O(n^\delta)$.

We will then have a quadratic approximation when $\theta \to 0$:

$$h_f(\rho_0(n)e^{i\theta}) - h_f(\rho_0(n)) =: -\frac{1}{2}\beta(\rho_0(n))\theta^2 + o(\theta^3) + O(n^\delta),$$

where $\beta(\rho(n))$ is a constant depending on $\rho(n)$, and the choice of $\delta$. We will then aim to determine an angle $\theta_0(n)$ such that

$$\beta(\rho_0(n))\theta_0(n)^2 \xrightarrow{\theta \to 0} +\infty, \quad h''(\rho_0(n))\theta_0(n)^3 \xrightarrow{\theta \to 0} 0.$$

For such a choice of $\theta_0(n)$, we deduce in particular that the behavior of the integral over the minor arc $C_0$ behaves like a Gaussian integral, whose bounds depend on $n$, which can be approximated.

Debruyne and Tennenbaum obtained a more general version of Meinardus' theorem in [DT20] by weakening condition iii) on the growth of $L_f$ when the imaginary part $\tau$ is large. Upon studying Witten's zeta functions, we will realize that Meinardus' theorem cannot be straightforwardly applied to obtain an asymptotic expression for the number of representations of the Lie algebra $r_g(n)$ that are of interest to us. However, by cleverly modifying the proof of Meinardus' Theorem, and then studying the residues of $\zeta_{sl(3)}(s)$ at $s = \frac{1}{2}, \frac{2}{3}$, and by calculating the special values $\zeta_{sl(3)}(0)$ and $\zeta'_{sl(3)}(0)$, Romik obtained the following theorem:

**Theorem 1.2.31.** [Rom17, Theorem 1.1] We have

$$r_{sl(3)}(n) \underset{n \to +\infty}{\sim} \frac{K}{n^{3/5}} \exp \left( A_1 n^{2/5} - A_2 n^{3/10} - A_3 n^{1/5} - A_4 n^{1/10} \right),$$

with the constant $A_1, A_2, A_3, A_4$ and $K$ being explicit:

Let

$$X = \left( \frac{1}{9} \Gamma \left( \frac{1}{3} \right)^2 \zeta \left( \frac{5}{3} \right) \right), \quad Y = -\sqrt{\pi} \zeta \left( \frac{1}{2} \right) \zeta \left( \frac{3}{2} \right).$$

The constants $A_1, A_2, A_3, A_4, K$ are such that:

$$A_1 = 5X^2 = 6.858260476...$$

$$A_2 = X^{-1}Y = 5.77360174...$$

$$A_3 = \frac{3}{80}X^{-4}Y^2 = 0.91134107...$$

$$A_4 = \frac{11}{3200}X^{-7}Y^3 = 0.35163754...$$

$$K = \frac{2\sqrt{3\pi}}{\sqrt{5}}X^{1/3} \exp\left(-\frac{1}{2560}X^{-10}Y^4\right) = 2.44629033486...$$

Note that in [Rom17], the computation of the constant $Y$ particularly depends on the special value $\zeta'_{sl(3)}(0) = -\ln(2\pi) + \ln(2)$ computed in [BD18]. Moreover, all these constants also depend on the calculation of the residues of $\zeta_{sl(3)}(s)$ at $s = \frac{1}{2}$ and $s = \frac{2}{3}$.

Later on, Bridges, Bringmann and Franke obtained in [BBF23] an asymptotic formula for the number of irreducible representations of dimension $n$ of sl(3) by studying the Witten zeta function $\zeta_{sl(3)}(s)$, and using a Tauberian type theorem.
Romik conjectures that it is possible to obtain similar results on the representation numbers $r_g(n)$ for other Lie algebras $g$. To extend this result, one needs to establish a Meinardus-type theorem, and also obtain information about the poles and residues of the corresponding Witten zeta function in the strip $0 \le \text{Re}(s) \le 1$ for the Witten zeta function $\zeta_g(s)$, and then obtain an expression for $\zeta'_g(0)$.

Motivated by this question, Bridges, Brindle, Bringmann, and Franke have established a more general variant of Meinardus' Theorem in the preprint [BBBF23]. Let $f: \mathbb{N} \to \mathbb{N}_0$. We define for all $q = e^{-z}$, $z \in H_0$, the functions

$$G_f(z) = \sum_{n \ge 0} p_f(n) q^n = \prod_{n \ge 1} \frac{1}{(1 - q^n)^{f(n)}}, \quad L_f(s) = \sum_{n \ge 1} \frac{f(n)}{n^s}.$$

Let $\Lambda := \mathbb{N}_0 \setminus f^{-1}(\{0\})$. We assume that:

(P1) Let $\alpha > 0$ be the largest poles of $L_f$. There exists an integer $L \in \mathbb{N}_0$ such that, for all prime numbers $p$,

$$|\Lambda \setminus (p\mathbb{N}_0 \cap \Lambda)| \ge L > \frac{\alpha}{2}.$$

(P2) There exists a real number $R \in \mathbb{R}_+$ such that $L_f$ is meromorphic on $\bar{H}_{-R} = \{z \in \mathbb{C}, \text{Re}(z) \ge -R\}$, and such that $L_f$ is holomorphic on the line $(\text{Re}(z) = -R)$. Furthermore, we assume that the meromorphic function $L_f^*(s) := \Gamma(s)\zeta(s+1)L_f(s)$ has only real poles $\alpha := \gamma_1 > \dots$, and that these poles are simple except at $s = 0$, where the pole may be a double pole.

(P3) There exists a real number $a < \frac{\pi}{2}$ such that, on each strip $\sigma_1 \le \sigma \le \sigma_2$ in the domain of holomorphy of $L_f$, we have

$$L_f(s) = O_{\sigma_1, \sigma_2} (e^{a|\tau|}) \quad (s = \sigma + i\tau),$$

as $|\tau| \to +\infty$.

**Theorem 1.2.32.** [BBBF23, Theorem 1.4] Assume conditions (P1), (P2), and (P3). Let $L$ be the real number from condition (P1) and $R$ be the real number from condition (P2). Then for some $M, N \in \mathbb{N}_0$,

$$p_f(n) = \lim_{n \to +\infty} \frac{C}{n^b} \exp \left( A_1 n^{\frac{\alpha}{\alpha+1}} + \sum_{j=2}^{M} A_j n^{\alpha_j} \right) \left( 1 + \sum_{j=2}^{N} \frac{B_j}{n^{\nu_j}} + O_{L,R} \left( n^{-\min\left\{\frac{2L-\alpha}{2(\alpha+1)}, \frac{R}{\alpha+1}\right\}} \right) \right),$$

with $0 \le \alpha_M < \dots < \alpha_1 := \frac{\alpha}{\alpha+1}$, $0 < \nu_2 < \dots < \nu_N$, $A_1, \dots, A_M$ and $B_2, \dots, B_N$ being explicit constants.

When the meromorphic function $L_f$ has only two strictly positive poles, Bridges, Brindle, Bringmann, and Franke obtained a more precise formula. Let us first introduce the following constants: Let $\alpha > \beta > 0$ be the only two positive poles of $L_f$. We denote $\omega_\alpha = \text{Res}_{s=\alpha}(L_f(s))$ and $\omega_\beta = \text{Res}_{s=\beta}(L_f(s))$ as the residues of $L_f$ at its positive poles, and we set $c_1 = \omega_\alpha\Gamma(\alpha+1)\zeta(\alpha+1)$, $c_2 = \omega_\beta\Gamma(\beta+1)\zeta(\beta+1)$, $c_3 = L_f(0)$. Let

$$C = \frac{e^{L_f'(0)} (\omega_\alpha \Gamma(\alpha+1) \zeta(\alpha+1))^{\frac{1-L_f(0)}{\alpha+1}}}{\sqrt{2\pi(\alpha+1)}} \quad b = \frac{1-L_f(0)+\frac{\alpha}{2}}{\alpha+1}.$$

Let $\mathcal{P}_R$ be the union of $\{0\}$ and of the poles greater than $-R$ of $L_f^*$

$$\mathcal{L} = \frac{1}{\alpha+1} \mathcal{P}_R + \sum_{\mu \in \mathcal{P}_R} \left( \frac{\mu+1}{\alpha+1} - 1 \right) \mathbb{N}_0,$$

$$\mathcal{M} := \frac{\alpha}{\alpha+1} \mathbb{N}_0 + \left( - \sum_{\mu \in \mathcal{P}_R} \left( \frac{\mu+1}{\alpha-1} - 1 \right) \mathbb{N}_0 \right) \cap \left[ 0, \frac{R+\alpha}{\alpha+1} \right].$$
**Theorem 1.2.33.** [BBBF23, Theorem 4.4] We assume conditions (P1), (P2), and (P3). Let L be the real number from condition (P1) and R be the real number from condition (P2). Moreover, we assume that $L_f$ has only two poles $\alpha > \beta > 0$ such that there exists an integer $l \in \mathbb{N}$ satisfying the inequality $\frac{l+1}{l}\beta < \alpha < \frac{l}{l-1}\beta$. Then we have

$$p_f(n) = \sum_{n \to +\infty} \frac{C}{n^b} \exp \left( A_1 n^{\frac{\alpha}{\alpha+1}} + A_2 n^{\frac{\beta}{\alpha+1}} + \sum_{j=3}^{l+1} A_j n^{\frac{(j-1)\beta}{\alpha+1} + \frac{j-2}{\alpha+1} + 2-j} \right) \left( 1 + \sum_{j=2}^{N} \frac{B_j}{n^{\nu_j}} + O_{L,R} \left( n^{-\min\left\{\frac{2L-\alpha}{2(\alpha+1)}, \frac{R}{\alpha+1}\right\}} \right) \right),$$

with $A_1 := (\omega_\alpha \Gamma(\alpha+1)\zeta(\alpha+1))^{\frac{1}{\alpha+1}} (1+\frac{1}{\alpha})$, and $A_2 := \frac{\omega_\beta \Gamma(\beta)\zeta(\beta+1)}{(w_\alpha \Gamma(\alpha)\zeta(\alpha+1))^{\frac{\beta}{\alpha+1}}}$, and for all $j \ge 3$,

$$A_j := K_j + \frac{c_1^{\frac{1}{\alpha+1}}}{\alpha} \sum_{m=1}^{l} \binom{-\alpha}{m} \sum_{\substack{0 \le k_1 \le \dots \le k_l \le m \\ |\mathbf{k}|=m \\ k_1+2k_2+\dots+l k_l=j-1}} \binom{m}{\mathbf{k}} \frac{K_2^{k_1} \dots K_{l+1}^{k_l}}{c_1^{\frac{m}{\alpha+1}}} \\ + \frac{c_2}{\beta c_1^{\frac{\beta}{\alpha+1}}} \sum_{m=1}^{l} \binom{-\beta}{m} \sum_{\substack{0 \le k_1 \le \dots \le k_l \le m \\ |\mathbf{k}|=m \\ k_1+2k_2+\dots+l k_l=j-2}} \binom{m}{\mathbf{k}} \frac{K_2^{k_1} \dots K_{l+1}^{k_l}}{c_1^{\frac{m}{\alpha+1}}}$$

with $(K_j)_{j \ge 3}$ described in [BBBF23, Lemma 4.3], and $0 < \nu_2 < \dots$ running over the positive elements of $\mathcal{L} + \mathcal{M}$.

**Remark 1.2.34.** To study the asymptotic behavior of $r_{g_2}(n)$, we would need to know $K_3$, and according to [BBBF23, Lemma 4.3], we have

$$K_3 = \frac{c_2^2(\alpha - 2\beta)}{2(\alpha + 1)^2 c_1^{\frac{2\beta+1}{\alpha+1}}}$$

with $c_1$ and $c_2$ being explicit constants such that

$$c_1 = \omega_\alpha \Gamma(\alpha + 1) \zeta(\alpha + 1), \quad c_2 = \omega_\beta \Gamma(\beta + 1) \zeta(\beta + 1).$$

Moreover, to study the asymptotic behavior of $r_{g_2}(n)$, we will see that the integer $l=2$ works because, we would then have $L_f(s) = \zeta_{g_2}(s)$, and the positive poles of this zeta function are $s = \frac{1}{3}$ and $s = \frac{1}{5}$.

By applying this theorem, the four authors obtained results similar to Romik for the number of representations $r_{g_0(5)}(n)$:

**Theorem 1.2.35.** [BBBF23, Theorem 1.3] For all positive integers $N \ge 1$, we have

$$r_{g_0(5)}(n) = \sum_{n \to +\infty} \frac{C}{n^{\frac{\alpha}{2}}} \exp \left( A_1 n^{\frac{1}{3}} + A_2 n^{\frac{2}{9}} + A_3 n^{\frac{1}{9}} + A_4 \right) \left( 1 + \sum_{j=2}^{N+1} \frac{B_j}{n^{\frac{j-1}{9}}} + O_N \left( n^{-\frac{N+1}{9}} \right) \right),$$

with

$$b = \frac{7}{12}, \quad C = \frac{e^{\zeta'_{g_0(5)}(0)} \Gamma(\frac{1}{4})^{\frac{1}{6}} \zeta(\frac{3}{2})^{\frac{1}{12}}}{2^{\frac{1}{3}} 3^{\frac{11}{24}} \sqrt{\pi}}, \quad A_1 = \frac{3^{\frac{4}{3}} \Gamma(\frac{1}{4})^{\frac{4}{3}} \zeta(\frac{3}{2})^{\frac{2}{3}}}{2^{\frac{8}{3}}}$$

$$A_2 = \frac{2^{\frac{8}{9}} \left(2^{\frac{1}{3}} + 1\right) \Gamma\left(\frac{1}{3}\right) \zeta\left(\frac{1}{3}\right) \zeta\left(\frac{4}{3}\right)}{3^{\frac{7}{9}} \Gamma\left(\frac{1}{4}\right)^{\frac{4}{9}} \zeta\left(\frac{3}{2}\right)^{\frac{2}{9}}}, \quad A_3 = \frac{2^{\frac{40}{9}} \left(2^{\frac{1}{3}} + 1\right)^2 \Gamma\left(\frac{1}{3}\right)^2 \zeta\left(\frac{1}{3}\right)^2 \zeta\left(\frac{4}{3}\right)^2}{3^{\frac{44}{9}} \Gamma\left(\frac{1}{4}\right)^{\frac{20}{9}} \zeta\left(\frac{3}{2}\right)^{\frac{10}{9}}}$$

$$A_4 = \frac{2^8 \left(2^{\frac{1}{3}} + 1\right)^3 \Gamma\left(\frac{1}{3}\right)^3 \zeta\left(\frac{1}{3}\right)^3 \zeta\left(\frac{4}{3}\right)^3}{3^8 \Gamma\left(\frac{1}{4}\right)^4 \zeta\left(\frac{3}{2}\right)^2}.$$
On the other hand, these four authors did not have an explicit formula for $\zeta'_{s_0(5)}(0)$. In our work, we will provide an expression for the special values $\zeta'_{s_0(5)}(0)$ and $\zeta'_{g_2}(0)$. We will also provide formulas for the residues of $\zeta_{s_0(5)}(s)$ at $s = \frac{1}{2}$ and $s = \frac{1}{3}$, and formulas for the residues of $\zeta_{g_2}(s)$ at $s = \frac{1}{3}$ and $s = \frac{1}{5}$. We will then be able to obtain an explicit asymptotic formula for the number of representations $r_{s_0(5)}(n)$ from the computation of $\zeta'_{s_0(5)}(0)$, allowing us to have an explicit expression for the coefficient $C$ from the previous theorem.

We should note that we will find a different value for $\zeta_{s_0(5)}(0)$ than in [BBBF23, Proposition 5.15]. We interpret this difference in results by suggesting that the authors may have made an error in replacing $\zeta(0)$ by $\frac{1}{2}$ instead of $-\frac{1}{2}$ in the proof of Proposition 5.15.

We will also obtain explicit formulas for $\zeta_{g_2}(0)$ and $\zeta'_{g_2}(0)$, and we will then be able to establish a theorem similar to the previous one, for the number of representations $r_{g_2}(n)$ using Theorem 1.2.33.

**Remark 1.2.36.** When we will apply the Theorem 1.2.32 to the number of representations $r_g(n)$ of a semi-simple Lie algebra, we will see that the coefficients $f(n)$ will correspond to the number of irreducible representations of dimension $n$ of $g$. Therefore, the function $L_f$ will correspond to the Witten zeta function $\zeta_g$.
1.3 First definitions

In this manuscript, we shall consider a family ($l_q$)$_{q \in [1, Q]}$ of linear forms of rank $P$ with coefficients in the Poincaré half-plane $H_0 = \{s = \sigma + i\tau \in \mathbb{C} | \sigma > 0\}$, where $P \ge 1$ and $Q \ge 1$ are positive integers. We also fix $\mathbf{d} = (d_p)_{i \in [1, P]}$, $\mathbf{d}' = (d'_q)_{q \in [1, Q]}$ as complex numbers in $H_0$ such that $d'_q = l_q(\mathbf{d})$ for all $1 \le q \le Q$.

**Notation.** Let $q \in [1, Q]$, we denote

$$l_q(\mathbf{n}) = \sum_{p=1}^{P} c_{q,p} n_p.$$

For the sake of clarity, throughout this manuscript, we will choose to use the index $p$ (or index $i$ if $p$ is already used) when summing or taking product over any subset $\mathcal{P}$ of $[1, P]$, and we will use the index $q$ (or the index $j$ if $q$ is already used) when summing or taking product over any subset $\mathcal{Q}$ of $[1, Q]$.

We will first define the incomplete gamma functions.

**Definition 1.3.1.** Let $\nu \in H_0$, $\theta > 0$, $s \in \mathbb{C}$. We call incomplete gamma functions the following functions:

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

These gamma functions are more general than those that can be found in the literature (for example, in [EMOT81, Chap.IX] or in [Cra12, §3]). The classical incomplete gamma functions can be recovered by setting $\nu = 1$. For our purposes, we will need an arbitrary $\nu$ in $H_0$.

We now introduce constants that will appear in theorem D. Through partial fraction decomposition, we can easily find:

**Proposition 1.3.2.** Let $(\mathbf{N}, \mathbf{N}') \in \mathbb{N}_0^P \times \mathbb{N}_0^Q$. Let's consider $\mathbb{K}$ the field above $\mathbb{Q}$, generated by the coefficients $c_{q,p}$, with $1 \le q \le Q$ and $1 \le p \le P$. Let $\mathcal{P} \subset [1, P]$, $j, f \in [1, Q]$, $\mathbf{v} = (\mathbf{v}_p)_{p \in \mathcal{P}} \in (\mathbb{N}_0^{[1,Q] \setminus \{j,f\}})^{\mathcal{P}}$, $\mathbf{w} = (\mathbf{w}_p)_{p \in \mathcal{P}^c} \in (\mathbb{N}_0^Q)^{\mathcal{P}^c}$. Let's note $\mathcal{P}^c = [1, P] \setminus \mathcal{P}$, $\mathbf{w}_p = (w_{p,q})_{q \in [1,Q]}$ and $\mathbf{v}_p = (v_{p,q})_{q \in [1,Q] \setminus \{j,f\}}$. There exist a polynomial $\tilde{P}_{\mathcal{P},j,f,\mathbf{v},\mathbf{w},\mathbf{N},\mathbf{N}'} \in \mathbb{K}[x]$ and two sequences of constants $(C_{\mathcal{P},j,f,\mathbf{v},\mathbf{w},\lambda}(\mathbf{N}, \mathbf{N}'))_{\lambda \in \mathbb{N}_0}$ and $(D_{\mathcal{P},j,f,\mathbf{v},\mathbf{w},(\lambda,p)}(\mathbf{N}, \mathbf{N}'))_{\lambda \in \mathbb{N}_0, p \in \mathcal{P}}$ such that all their terms belong in the field $\mathbb{K}$, and such that

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

We consider $P_{\mathcal{P},j,f,\mathbf{v},\mathbf{w},\mathbf{N},\mathbf{N}'}$ the primitive function which vanish at 1 of the function $\tilde{P}_{\mathcal{P},j,f,\mathbf{v},\mathbf{w},\mathbf{N},\mathbf{N}'}$. Let's note $E_{\mathcal{P},j,f,\mathbf{v},\mathbf{w}}(\mathbf{N}, \mathbf{N}') \in \mathbb{K}$ the constant term of $P_{\mathcal{P},j,f,\mathbf{v},\mathbf{w},\mathbf{N},\mathbf{N}'}$.

We now set

$$F_{\mathcal{P},j,f,\mathbf{v},\mathbf{w}}(\mathbf{N}, \mathbf{N}') = -E_{\mathcal{P},j,f,\mathbf{v},\mathbf{w}}(\mathbf{N}, \mathbf{N}') - \sum_{\lambda=2}^{N'_f+1-\mathbf{w}(f)} \frac{C_{\mathcal{P},j,f,\mathbf{v},\mathbf{w},\lambda}(\mathbf{N}, \mathbf{N}')}{\lambda-1} \\ 
- \frac{1}{c_{f,p}} \sum_{p \in \mathcal{P}} \sum_{\lambda=2}^{(|\mathbb{N}|_{\mathcal{P}} + |\mathcal{P}| + |\mathbf{v}|)} \frac{D_{\mathcal{P},j,f,\mathbf{v},\mathbf{w},(\lambda,p)}(\mathbf{N}, \mathbf{N}')}{\lambda-1} ((c_{j,p} + c_{f,p})^{1-\lambda} - c_{j,p}^{1-\lambda}) \\ 
+ \frac{1}{c_{f,p}} \sum_{p \in \mathcal{P}} D_{\mathcal{P},j,f,\mathbf{v},\mathbf{w},(1,p)}(\mathbf{N}, \mathbf{N}') \ln \left(1 + \frac{c_{f,p}}{c_{j,p}}\right).$$
**Remark 1.3.3.** 1) Note that if $-N'_f - 1 + \mathbf{w}(f) \ge 0$, then $C_{\mathcal{P},j,f,v,w,\lambda} = 0$ for all $\lambda$.

2) In the product $\prod_{p \in \mathcal{P}} (c_{j,p} + c_{f,p}x)^{(-N_p-1-|v_p|)}$, some terms of the product may appear multiple times.

This is why the sum $\sum_{\lambda=1}^{(|N|_P+|\mathcal{P}|+|v|)}$ goes up to $|N|_P + |\mathcal{P}| + |v|$. If this occurs, then we have multiple possible choices for the constants $D_{\mathcal{P},j,f,v,w,(\lambda,p)}$. Note that these different choices will have no impact on the definition of the constant $F_{\mathcal{P},j,f,v,w}(\mathbf{N}, \mathbf{N}')$.

3) The constant $F_{\mathcal{P},j,f,v,w}(\mathbf{N}, \mathbf{N}')$ will appear in the calculation of an integral, in Lemma 3.1.4.

**Example 1.3.4.** Let $\mathcal{P} = \emptyset$, we consider for all $\lambda \ge 1$,

$$C_{\emptyset,j,f,v,w,\lambda}(\mathbf{N}, \mathbf{N}') = \delta_{\lambda,N'_f+1-\mathbf{w}(f)}$$

$$D_{\emptyset,j,f,v,w,(\lambda,p)}(\mathbf{N}, \mathbf{N}') = 0$$

$$E_{\emptyset,j,f,v,w}(\mathbf{N}, \mathbf{N}') = 0 \text{ if } \mathbf{w}(f) \le N'_f$$

$$E_{\emptyset,j,f,v,w}(\mathbf{N}, \mathbf{N}') = -\frac{1}{\mathbf{w}(f) - N'_f} \text{ if } \mathbf{w}(f) > N'_f,$$

where we noted $\delta_{\lambda, \mathbf{w}(f)}$ the Kronecker symbol.

In particular, we get that $F_{\emptyset,j,f,0,0}(0,0) = 0$.

**Example 1.3.5.** Let $\mathcal{P} = \{p\}$, $j \ne f \in [1, Q]$, $(\mathbf{N}, \mathbf{N}') = (0,0)$, $v = 0$ and $\mathbf{w} = e_f$, we then get that

$$x_f^{-N'_f-1+\mathbf{w}(f)}(c_{j,p} + c_{f,p}x_f)^{-N_p-1-|v_p|} = (c_{j,p} + c_{f,p}x_f)^{-1}.$$

Therefore, we have $F_{\{p\},j,f,0,e_f} = \frac{1}{c_{f,p}} \ln \left( 1 + \frac{c_{f,p}}{c_{j,p}} \right)$.

**Example 1.3.6.** Let $\mathcal{P} = \{p\}$, $j \ne f \in [1, Q]$, $(\mathbf{N}, \mathbf{N}') = (0,0)$, $v = 0$ and $\mathbf{w} = e_j$, we then get that

$$x_f^{-N'_f-1+\mathbf{w}(f)}(c_{j,p} + c_{f,p}x_f)^{-N_p-1-|v_p|} = x_f^{-1}(c_{j,p} + c_{f,p}x_f)^{-1} = \frac{1}{c_{j,p}}x^{-1} - \frac{c_{f,p}}{c_{j,p}}(c_{j,p} + c_{f,p}x_f)^{-1}.$$

Therefore, we have $F_{\{p\},j,f,0,e_j} = -\frac{1}{c_{j,p}} \ln \left( 1 + \frac{c_{f,p}}{c_{j,p}} \right)$.

We shall study in our thesis the following multizeta function:

**Definition 1.3.7.** Let $(\mathbf{s}, \mathbf{s}') \in H_1^P \times H_1^Q$, then we set

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

**Remark 1.3.8.** This multizeta function is of the multiple Hurwitz-Lerch type described in Definition 1.2.24, without torsion:

$$Z(\mathbf{s}, \mathbf{s}') = \zeta^{HL}(0, \tilde{\mathbf{d}}, \tilde{\mathbf{c}}, (s_1, \dots, s_P, s'_1, \dots, s'_Q)) = \sum_{\mathbf{n} \in \mathbb{N}_0^P} \frac{1}{\prod_{q=1}^{\tilde{Q}} (\tilde{d}_q + \sum_{p=1}^{P} \tilde{c}_{q,p} n_p)^{s_q}},$$

with $P_0 = P$, $\tilde{Q} := P + Q$, $\mathcal{B}^c = \cup_{1 \le p \le P} [1, P] \setminus \{p\} \times \{p\} \subset [1, Q + P] \times [1, P]$, $\tilde{\mathbf{d}} = (\mathbf{d}, \mathbf{d}')$, and with $\tilde{\mathbf{c}} = (c_{(q,p)})(q,p) \in [1, Q'] \times [1, P]$ such that

$$\left\{ \begin{array}{l} \tilde{c}_{q,p} = 0 \text{ if } q \ne p \text{ and } q \le P \\ \tilde{c}_{p,p} = 1 \text{ if } p \in [1, P] \\ \tilde{c}_{q,p} = c_{q-P,p} \text{ if } q > P. \end{array} \right.$$

Therefore, we can apply the Theorem 1.2.25 for $Z(\mathbf{s}, \mathbf{s}')$.
With the help of the previous remark, and Theorem 1.2.25, we see that the multizeta function $Z(s, s')$ is meromorphic over the entire $\mathbb{C}^P \times \mathbb{C}^Q$, and this theorem also provides information about the set of singularities of $Z(s, s')$. We also have a domain of holomorphy, however for our needs, we require a larger domain of convergence than what is provided by Theorem 1.2.25. We will then show a more precise domain of convergence in the following proposition:

**Proposition 1.3.9.** We set

$$ \mathcal{D} = \mathcal{D}_{P,Q} := \left\{ (\mathbf{s}, \mathbf{s}') \in \mathbb{C}^P \times \mathbb{C}^Q \mid \sum_{p \in \mathcal{P}} \sigma_p + \sigma'_1 + \dots + \sigma'_Q > |\mathcal{P}|, \emptyset \neq \mathcal{P} \subset \llbracket 1, P \rrbracket \right\}. $$

The multizeta function $Z(s, s')$ has a domain of convergence included in $\mathcal{D}$, and is meromorphic over $\mathbb{C}^P \times \mathbb{C}^Q$, with singularities included in the following union of hyperplanes:

$$ \left\{ \begin{array}{l} \sum_{p \in \mathcal{P}} s_p + \sum_{q \in \mathcal{Q}} s'_q = Q + P - n \quad (P \subset \llbracket 1, P \rrbracket, \mathcal{Q} \subset \llbracket 1, Q \rrbracket, |\mathcal{P} \cup \mathcal{Q}| \ge 2, n \in \mathbb{N}_0) \\ s_p = 1, \dots, P \quad (1 \le p \le P) \\ s'_q = 1, \dots, P \quad (1 \le q \le Q). \end{array} \right. $$

We will denote in the same way $Z(s, s')$ its meromorphic extension. In general, the values at nonpositive tuple do not make sense since some nonpositive tuple are singularities for $Z$. That's why we will fix a direction $(\mu, \mu')$ and a $P+Q$-tuple $-(\mathbf{N}, \mathbf{N}')$, and study the multizeta function $Z$ at this $P+Q$-tuple, and in the chosen direction. It follows from Proposition 1.3.9 that $s \mapsto Z(-\mathbf{N} + s\mu, -\mathbf{N}' + s\mu')$ is meromorphic on $\mathbb{C}$, and is holomorphic for $s = \sigma + i\tau$, with $\sigma$ sufficiently large. We can then define the following zeta function:

**Definition 1.3.10.** Let $(\mathbf{N}, \mathbf{N}') \in \mathbb{N}_0^P \times \mathbb{N}_0^Q$, and $(\mu, \mu') \in (\mathbb{R}_+)^P \times (\mathbb{R}_+^*)^Q$. We then define for $s \in \mathbb{C}$,

$$ Z_{\mathbf{N},\mathbf{N}'}(s) := Z(-\mathbf{N} + s\mu, -\mathbf{N}' + s\mu'). $$

We also consider the following univariate function, which consists of evaluating the previously described function in the direction $(\mu, \mu') = (\mathbf{1}, \mathbf{1})$, and for $(\mathbf{N}, \mathbf{N}') = (\mathbf{0}, \mathbf{0})$.

**Definition 1.3.11.** We also define for $s \in \mathbb{C}$ the following function

$$ Z_{\Delta} : s \in \mathbb{C} \mapsto Z((s, \dots, s), (s, \dots, s)). $$

**Remark 1.3.12.** If we set $\mu = \mathbf{0}$, and $\mu' = \mathbf{1}$, then $Z_{\mathbf{0},\mathbf{0}'}(s)$ corresponds to a Shintani-type zeta function

$$ Z_{\mathbf{0},\mathbf{0}}(s) = \sum_{\mathbf{n} \in \mathbb{N}_0^P} \prod_{q=1}^Q (l_q(\mathbf{n}) + d'_q)^{-s}. $$

Keep in mind that the coefficients $d'_q$ are not free, and depend on the choices of $d_p$ and on the choices of the coefficients $(c_{p,q})_{1 \le p \le P, 1 \le q \le Q}$.

By Proposition 1.3.9, we know that $Z_{\mathbf{N},\mathbf{N}'}$ is holomorphic on $\{s \in \mathbb{C}|(-\mathbf{N} + \mu s, -\mathbf{N}' + \mu' s) \in \mathcal{D}\}$. Therefore, there is a real $\sigma_{0,\mathbf{N},\mathbf{N}'}$ such that the function $Z_{\mathbf{N},\mathbf{N}'}$ is holomorphic on $H_{\sigma_{0,\mathbf{N},\mathbf{N}'}}$. We shall say that $Z_{\mathbf{N},\mathbf{N}'}$ is holomorphic on $\sigma \gg_{\mathbf{N},\mathbf{N}'} 1$, or sometimes only $\sigma \gg 1$ when there is no ambiguity.

In order to give meaning to the directional values of $Z$ at nonpositive tuple, we must prove that $Z_{\mathbf{N},\mathbf{N}'}$ is regular at $s = 0$. Works by Komori indeed prove that this directional value has a meaning.

In this manuscript, we will detail a different extension method in Proposition 2.2.4 to prove that these directional values have meaning. That proposition will **mainly** allow us to obtain explicit formulas for the special directional values of $Z$, as well as those of its derivative.
**Notation.** 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 consider the values in $(-\mathbf{N}, -\mathbf{N}')$ of $Z(\mathbf{s}, \mathbf{s}')$ along the direction $(\boldsymbol{\mu}, \boldsymbol{\mu}')$,

$$Z(-(\mathbf{N}, \mathbf{N}')) := Z_{\mathbf{N}, \mathbf{N}'}(0),$$
$$\underset{\boldsymbol{\mu}, \boldsymbol{\mu}'}{Z'(-(\mathbf{N}, \mathbf{N}'))} := \partial_s(Z_{\mathbf{N}, \mathbf{N}'}(s))|_{s=0}.$$

and we consider the derivative value along a direction $(-\mathbf{N}, -\mathbf{N}')$ of $Z(\mathbf{s}, \mathbf{s}')$, along the direction $(\boldsymbol{\mu}, \boldsymbol{\mu}')$,

$$Z'(-(\mathbf{N}, \mathbf{N}')) := \partial_s(Z_{\mathbf{N}, \mathbf{N}'}(s))|_{s=0}.$$

We will now introduce auxiliary functions (which we can call generalized Barnes zeta functions), for the purpose of studying the derivative values along a direction $Z'(-(\mathbf{N}, \mathbf{N}'))$. These functions will appear in the formulas of such values.

**Definition 1.3.13.** Let $\mathbf{R} \in \mathbb{N}_0^P$, and $j \in [1, Q]$. We consider for all $\sigma \gg 1$,

$$\varphi_{\mathbf{R}}^j(s) = \sum_{\mathbf{n} \in \mathbb{N}_0^P} (n_1 + d_1)^{R_1} \dots (n_P + d_P)^{R_P} (l_j(\mathbf{n}) + d_j')^{-s}.$$

**Remark 1.3.14.** If $\mathbf{R} = \mathbf{0}$, we obtain the classical Barnes zeta function $\varphi_0^j(s) = \zeta_P^B(s, d_j'|c_{j,1}, \dots, c_{j,P})$.

The function $\varphi_{\mathbf{R}}^j(s)$ corresponds to $Z_{\mathbf{N}, \mathbf{N}'}(s)$, with $Q = 1$, $\boldsymbol{\mu} = \mathbf{0}$, $\boldsymbol{\mu}' = 1$, $\mathbf{N} = \mathbf{R}$ and $\mathbf{N}' = 0$. Particularly, we know that it is meromorphic on $\mathbb{C}$, and is regular at nonpositive integers. The first derivative of these auxiliary functions will appear in the formulas of $Z'(-(\mathbf{N}, \mathbf{N}'))$. Therefore, it is crucial to have an explicit formula for the derivative of such functions.

## 1.4 Main results

As mentioned in the previous section, we will prove, using another method, the result of Komori on the existence of the directional values $Z(-(\mathbf{N}, \mathbf{N}'))$ in Proposition 2.2.4. To obtain this result, we use in Chapter 2 a strategy called Crandall's expansion, developed in [Cra12], which was initially used to obtain numerical approximations of certain zeta functions and certain $L$ functions. This strategy was subsequently used by Borwein and Dilcher in [BD18] to study the values of a Mordell-Tornheim type zeta function, and to approximate its derivative at nonpositive integers. Subsequently, Onodera provided exact formulas for the values at nonpositive integers of a generalized Mordell-Tornheim type zeta function, as well as the values of its first and second order derivatives in [Ono21]. The trick used by Onodera to compute the values of the derivatives at nonpositive integers partly relies on introducing auxiliary functions resembling Barnes' zeta functions, with a polynomial term in the numerator of the general term of the series, and then providing a formula for the derivatives of these auxiliary functions.

More precisely, we shall use a we shall describe an analytic continuation formula for the meromorphic function $Z_{\mathbf{N}, \mathbf{N}'}$ in Proposition 2.2.4:

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

where $K_{\mathbf{N}, \mathbf{N}'}$ is holomorphic on $\mathbb{C}$, and vanish at $s = 0$, and $J_{\mathbf{N}, \mathbf{N}'}(\theta, s)$ is meromorphic on $\mathbb{C}$, with poles that are all simple, of the form

$$s = \frac{n}{\mu_p} \qquad (1 \le p \le P \text{ such that } \mu_p \ne 0, n \in \mathbb{Z}_{\ge N_p+1})$$
$$s = \frac{n}{|\boldsymbol{\mu}'| + |\boldsymbol{\mu}|_P} \qquad (\emptyset \ne \mathcal{P} \subset [1, P], n \in \mathbb{Z}^*_{\le |\mathbf{N}|_P + |\mathbf{N}'| + |\mathcal{P}|}).$$
In particular, we will see that 0 isn't a pole for the meromorphic function $Z_{\mathbf{N},\mathbf{N}'}$, therefore obtaining that directional values of the function $Z(\mathbf{s}, \mathbf{s}')$ have a meaning.

The fact that $Z_{\mathbf{N},\mathbf{N}'}$ is meromorphic extension over $\mathbb{C}$ directly follows from Proposition 1.3.9, and this same theorem also provides candidates for the poles of this function. Indeed, it is sufficient to replace $s_p$ by $-N_p + \mu_p s$ and $s'_q$ by $-N'_q + \mu'_q s$ to obtain candidate poles. However, the set of poles provided here is more precise.

Let's note that the term $K_{\mathbf{N},\mathbf{N}'}(\theta, s)$ will not contribute to the formula for the special directional values, as it vanishes at $s=0$. On the other hand, if we differentiate this term, it is non-zero in general, and is pathological. We will therefore use the auxiliary functions $(\varphi_{\mathbf{R}}^j)_{j \in [1,Q]}$ that will help retrieve the information of the term $\partial_s(K_{\mathbf{N},\mathbf{N}'}(\theta, s))$ at $s=0$. We will then be able to evaluate the extension formula at $s=0$ to obtain an explicit formula for the directional values of the derivatives.

**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

$$Z(-(\mathbf{N}, \mathbf{N}')) = (-1)^{|\mathbf{N}'|} \sum_{\mathcal{P} \subseteq [1, P]} (-1)^{|\mathbf{N}|_{\mathcal{P}} + |\mathcal{P}|} \left( \prod_{p \in \mathcal{P}} N_p! \right) \sum_{\substack{\mathbf{k}=(k_p)_{p \in \mathcal{P}} \in \mathbb{N}_0^{\mathcal{P}^c} \\ |\mathbf{k}| = |\mathbf{N}'| + |\mathbf{N}|_{\mathcal{P}} + |\mathcal{P}|}} \sum_{j=1}^{Q} \frac{(-1)^{N'_j} \mu'_j N'_j!}{|\mu'| + |\mu|_{\mathcal{P}}} Q_{\mathbf{N}, \mathbf{N}'}^0(j, \mathcal{P}, \mathbf{k}) \quad (1.8)$$

$$\prod_{p \in \mathcal{P}^c} \frac{\zeta(-N_p - k_p, d_p)}{k_p!},$$

with

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

where we noted

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

and

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

**Remark 1.4.1.** 1) The formula describing $Q_{\mathbf{N},\mathbf{N}'}^0(j, \mathcal{P}, \mathbf{k})$ is a finite sum. Indeed, we see that

$$\sum_{\substack{q=1 \\ q \neq j}}^{Q} (N'_q - \mathbf{w}(q)) = \sum_{\substack{q=1 \\ q \neq j}}^{Q} \mathbf{v}(q) = \sum_{(p,q) \in \mathcal{P} \times ([1,Q] \setminus \{j\})} v_{p,q}.$$

Since $\mathbf{w}(q)$ is nonnegative for all $q \in [1, Q]$, we deduce the following upper bound:

$$\sum_{(p,q) \in \mathcal{P} \times ([1,Q] \setminus \{j\})} v_{p,q} \leq |\mathbf{N}'|.$$

Thus, for all $p \in \mathcal{P}$ and all $q \in [1, Q] \setminus \{j\}$, we have $0 \leq v_{p,q} \leq |\mathbf{N}'|$.
2) The expression for the values of $Z_{\mu,\mu'}(\mathbf{N}, \mathbf{N}')$ is described by a finite sum. Indeed, for all $\mathbf{k} \in \mathbb{N}_0^{\mathcal{P}^c}$ such that $|\mathbf{k}| = |\mathbf{N}'| + |\mathbf{N}_{||\mathcal{P}}| + |\mathcal{P}|$ we have $0 \le k_p \le \mathbf{N}'|\mathbf{N}_{||\mathcal{P}}| + |\mathcal{P}|$ for all $p \in \mathcal{P}^c$.

3) Terms of the form $\zeta(-n, d)$ with $n \in \mathbb{N}_0$ and $d \in H_0$ are expressed using Bernoulli polynomials:

$$ \zeta(-n, d) = -\frac{B_{n+1}(d)}{n+1}. $$

Moreover, at $d=1$, we have $\zeta(-n, 1) = \zeta(-n)$, and

$$ \zeta(-n) = (-1)^n \frac{b_{n+1}}{n+1}, $$

where $b_{n+1}$ is the $n+1$-th Bernoulli number.

4) If all the coefficients $d_p$ are equal to 1, then the product $\prod_{p \in \mathcal{P}^c} \frac{\zeta(-N_p - k_p, d_p)}{k_p!}$ will often be zero. Indeed, if there exists $p \in \mathcal{P}^c$ such that $N_p + k_p$ is even and greater than 2, then the product is zero.

Taking $(\mathbf{N}, \mathbf{N}') = (\mathbf{0}, \mathbf{0})$, we obtain the following simplified expressions:

**Corollary A1.** Let $(\mu, \mu') \in \mathbb{R}_+^P \times \mathbb{R}_*^{Q^c}$. We have

$$ Z(-\mathbf{0}, \mathbf{0}) = \sum_{\mathcal{P} \subseteq [1, P]} (-1)^{|\mathcal{P}|} \sum_{\substack{\mathbf{k}=(k_p)_{p \in \mathcal{P}^c} \in \mathbb{N}_0^{\mathcal{P}^c} \\ |\mathbf{k}| = |\mathcal{P}|}} \sum_{j=1}^{Q} \frac{\mu_j'}{|\mathbf{\mu}'| + |\mathbf{\mu}|_{\mathcal{P}}}|\mathbf{\mu}'| + |\mathbf{\mu}|_{\mathcal{P}}|\mathbf{\mu}|} \prod_{p \in \mathcal{P}^c} c_{j,p}^{k_p} \frac{\zeta(-k_p, d_p)}{k_p!}. \quad (1.9) $$

**Remark 1.4.2.** We get the same formula as [Ono21] in $N = 0$, after replacing the direction with $1 = (1, ..., 1)$, and taking $Q = 1$.

Via Theorem A, we also get a formula for $Z(\mathbf{s}, \mathbf{s}')$ along the diagonal by taking the direction $(\mu, \mu') = (\mathbf{1}, \mathbf{1})$:

**Corollary A2.** Let $N \in \mathbb{N}_0$. We have

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

with

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

where we noted

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

and

$$ \forall q \in [1, Q] \setminus \{j\}, \quad \mathbf{v}(q) = \sum_{p \in \mathcal{P}} v_{p,q}. $$
**Remark 1.4.3.** If we take $Q = 1$, $P \ge 1$ a positive integer, and $l_1(\mathbf{n}) = n_1 + \dots + n_P$, we find via the previous corollary that $Q_N^0(\mathcal{P}, \mathbf{k}) = 1$, and that

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

This formula matches the one obtained by Onodera in [Ono21, Theorem 3].

We also can formulate a qualitative version of the Theorem A:

**Corollary A3.** Let $(\boldsymbol{\mu}, \boldsymbol{\mu}') \in \mathbb{R}_+^P \times \mathbb{R}_+^Q$, and $(\mathbf{N}, \mathbf{N}') \in \mathbb{N}_0^P \times \mathbb{N}_0^Q$. We consider $\mathbb{K}$ the field above $\mathbb{Q}$ generated by the coefficients $(d_p)_{p \in [1, P]}$ and $(c_{q,p})_{p \in [1, P], q \in [1, Q]}$, and by the directions $\mu_p$ for $p \in [1, P]$, and by the coefficients $\mu'_q$ for $q \in [1, Q]$. We then have $Z(-(\mathbf{N}, \mathbf{N}')) \in \mathbb{K}_{\mu, \mu'}$.

We also have an expression for the directional derivative at nonpositive tuple of $Z$. This expression involves "auxiliary" functions, which we will call generalized Barnes-type zeta functions.

If we assume that a linear form $l_j$ has rational coefficients, we have a relationship between the auxiliary function $\varphi_{\mathbb{R}}^j(s)$ and the Hurwitz zeta function:

**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 $\mathbb{R} \in \mathbb{N}_0^P$, we have for all $s \in \mathbb{C} \setminus \mathbb{N}$,

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

Deriving with respect to $s$ the previous equality, and evaluating that derivative at $s = -N$ with $N$ being a nonnegative integer, we then found the following corollary:

**Corollary B1.** With the same notations as in Proposition B, we then get for all $\mathbb{R} \in \mathbb{N}_0^P$,

$$\varphi_{\mathbb{R}}^{j'}(-N) = x_j(\mathbf{c})^N \prod_{p=1}^{P} \beta_{j,p}(\mathbf{c})^{R_p} \sum_{\substack{0 \le v_1 \le \beta_{j,1}(\mathbf{c})-1 \\ 0 \le v_p \le \beta_{j,P}(\mathbf{c})-1}} \sum_{\mathcal{P} \subseteq [1, P]} \left( \prod_{p \in \mathcal{P}^c} R_p! \right) \cdot \sum_{\substack{\mathbf{k} \in \mathbb{N}_0^P, k' \in \mathbb{N}_0 \\ |\mathbf{k}|+k'=|\mathbb{R}|^{|\mathcal{P}^c|+|\mathcal{P}^c|-1}}} (-1)^{|\mathbf{k}|} \prod_{p \in \mathcal{P}} \frac{\zeta\left(-R_p - k_p, \frac{d_p+v_p}{\beta_{j,p}(\mathbf{c})}\right)}{k_p!} \cdot \left( \frac{\zeta'(-N-k', l_j(x_j(\mathbf{c}))^{-1}(\mathbf{d}+\mathbf{v})))}{k'!} - \ln(x_j(\mathbf{c})) \frac{\zeta(-N-k', l_j(x_j(\mathbf{c}))^{-1}(\mathbf{d}+\mathbf{v})))}{k'!} \right).$$

We now can give a qualitative version of our result on special values of the directional derivative of $Z(\mathbf{s}, \mathbf{s}')$ at nonpositive integers:
**Theorem C.** 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 consider $\mathbb{K}$ the field above $\mathbb{Q}$ generated by the coefficients $(d_p)_{p \in [1, P]}$ and $(c_{q,p})_{p \in [1, P], q \in [1, Q]}$, and by the directions $(\mu_p)_{p \in [1, P]}$, $(\mu'_q)_{q \in [1, Q]}$. Then,

$$Z'(-(\mathbf{N}, \mathbf{N}')) \in \mathrm{Vect}_{\mathbb{K}} \left[ \gamma, \ln \left( 1 + \frac{c_{q,p}}{c_{j,p}} \right)_{1 \le q, j \le Q, 1 \le p \le P}, \left( \zeta'(-n, d_p)_{n \in \mathcal{P}, |\mathbf{N}|+|\mathbf{N}'|+P}, \right. \right. \\ \left. \left. (\varphi_{\mathbf{n}}^j(-N'_j))_{\mathbf{n} \in [0, |\mathbf{N}|+|\mathbf{N}'|]^P, j \in [1, Q]} \right] ,$$

where $\gamma$ is Euler's constant, and $\varphi_j$ are the zeta function of generalized Barnes type described in Definition 1.3.13.

If the linear forms $(l_j)_{j \in [1, Q]}$ are all rationals, then we can use Corollary B1 in order to obtain a more precise result:

**Corollary C1.** 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 assume that the linear forms $(l_j)_{j \in [1, Q]}$ and the coefficients $(d_p)_{p \in [1, P]}$ are rationals. We consider the field $\mathbb{K}$ above $\mathbb{Q}$ generated by the directions $(\mu_p)_{p \in [1, P]}$, $(\mu'_q)_{q \in [1, Q]}$. For all $1 \le p \le P$, we write $c_{q,p} = \frac{a_{q,p}}{b_{q,p}} \in \mathbb{Q}$ the irreducible decomposition of $c_{q,p}$ with $a_{q,p} > 0$ and $b_{q,p} > 0$. We set $x_q(\mathbf{c}) := \frac{\mathrm{lcm}(a_{q,p})}{\mathrm{gcd}(b_{q,p})} \in \mathbb{Q}_+^*$, and $\beta_{q,p}(\mathbf{c}) := \frac{x_q(\mathbf{c})}{c_{q,p}} \in \mathbb{N}$ for all $1 \le p \le P$. Let's also consider the following set

$$A := \{l_j(x_c^{-1}(\mathbf{d} + \mathbf{v})) | j \in [1, Q], \mathbf{v} \in [0, \beta_{j,1} - 1] \times \dots \times [0, \beta_{j,P} - 1]\}.$$

We then have that

$$Z'(-(\mathbf{N}, \mathbf{N}')) \in \mathrm{Vect}_{\mathbb{K}} \left[ \gamma, \ln \left( 1 + \frac{c_{q,p}}{c_{j,p}} \right)_{q,j \in [1, Q]}, (\ln(x_j(\mathbf{c})))_{1 \le j \le Q}, (\zeta'(-n, y))_{0 \le n \le P} \right]$$

where $\gamma$ is Euler's constant.

The formula for $Z'(-(\mathbf{N}, \mathbf{N}'))$ is actually explicit. It has a formula involving the special values of the derivatives of the generalized Barnes zeta functions $\varphi_j^R$, special values of the derivatives with respect to $s$ of the Hurwitz zeta function, and special values of the Hurwitz zeta function itself. We detail this formula in the theorem below.
**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

$$Z'(-(\mathbf{N}, \mathbf{N}')) = \tag{1.10}$$

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

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

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

$$- \sum_{j=1}^{Q} (\gamma - h_{N'_j}) \mu'_j N'_j! \sum_{\substack{\mathbf{u}=(\mathbf{u}_q)_{q \in [1,Q]\setminus\{j\}} \in (\mathbb{N}_0^P)^{[1,Q]\setminus\{j\}} \\ \forall q \in [1,Q]\setminus\{j\}, |\mathbf{u}_q|=N'_q}} \left( \left( \prod_{q=1}^{Q} \binom{N'_q}{\mathbf{u}_q} \prod_{p=1}^{P} c_{q,p}^{u_{q,p}} \right) \partial_s \varphi_{\mathbf{R}(\mathbf{N},j,\mathbf{u})}^j(s) \right)$$

$$\cdot \left( \sum_{\mathcal{P} \subseteq [1, P]} (-1)^{|\mathbf{R}(\mathbf{N}, j, \mathbf{u})|_{\mathcal{P}} + |\mathcal{P}|} \prod_{p \in \mathcal{P}} c_{j,p}^{-R(\mathbf{N}, j, \mathbf{u})_p - 1} R(\mathbf{N}, j, \mathbf{u})_p! \sum_{\substack{\mathbf{k}=(k_p)_{p \in \mathcal{P}} \in \mathbb{N}_0^{\mathcal{P}^c} \\ |\mathbf{k}|=N'_j + |\mathbf{R}(\mathbf{N}, j, \mathbf{u})|_{\mathcal{P}} + |\mathcal{P}|}} \prod_{p \in \mathcal{P}} c_{j,p}^{k_p} \frac{\zeta(-\mathbf{R}(\mathbf{N}, j, \mathbf{u})_p - k_p, d_p)}{k_p!} \right),$$

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 $j \in [1, Q]$,

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

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

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

**Remark 1.4.4.** This theorem thus reduces the computation of directional derivatives at the nonpositive $(Q+P)$-tuple $Z'(-(\mathbf{N}, \mathbf{N}'))$ to the computations of values at nonpositive integers of the derivative of the auxiliary functions

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

If the coefficients of the linear forms $l_j$ are rationals, then we have an explicit expression for the value of
$\partial_s \varphi_{\mathbf{R}}^j(-N)$ in terms of the special values of the derivatives of the Hurwitz zeta function via Proposition
B.

**Remark 1.4.5.** We observe that the formula for $Z'(-(\mathbf{N}, \mathbf{N}'))$ involves only finite sums. Indeed, it is
$\mathbf{\mu}, \mathbf{\mu}'$

clear that the sum $\sum_{\mathbf{k}=(k_p)_{p \in \mathcal{P}} \in \mathbb{N}_0^{\mathcal{P}^c}} |\mathbf{k}|=|\mathbf{N}'|+|\mathbf{N}'| + |\mathcal{P}|$ is finite. On top of that the sum

$$
\sum_{\substack{\mathbf{w}=(\mathbf{w}_p)_{p \in \mathcal{P}} \in (\mathbb{N}_0^Q)^{\mathcal{P}^c} \\ \forall p \in \mathcal{P}^c |\mathbf{w}_p|=k_p}} \sum_{\substack{\mathbf{v}=(\mathbf{v}_p)_{p \in \mathcal{P}} \in (\mathbb{N}_0^{[1,Q]} \setminus \{j\})^{\mathcal{P}} \\ \forall q \neq j, \mathbf{v}(q)+\mathbf{w}(q)=N'_q}} \text{in the formula of } Q_{\mathbf{N},\mathbf{N}'}^1 \text{ and of } Q_{\mathbf{N},\mathbf{N}'}^0(j, \mathcal{P}, \mathbf{k}) \text{ is also finite} \\
\text{because}
$$

$$
\begin{align*}
\forall a \in \mathcal{P}, \forall b \in [1, Q] \setminus \{j\}, \quad 0 \le v_{a,b} &\le \sum_{\substack{q=1 \\ q \neq j}}^{Q} \mathbf{v}(q) + \mathbf{w}(q) = |\mathbf{N}'| \\
\forall p \in \mathcal{P}^c, \forall q \in [1, Q], \quad 0 \le w_{p,q} &\le |\mathbf{w}_p| = k_p \le |\mathbf{k}| = |\mathbf{N}'| + |\mathbf{N}'|_{\mathcal{P}^c} + |\mathcal{P}|.
\end{align*}
$$

For the same reasons, the sum $\sum_{\mathbf{w}=(\mathbf{w}_p)_{p \in \mathcal{P}} \in (\mathbb{N}_0^Q)^{\mathcal{P}^c}} \sum_{\substack{\mathbf{v}=(\mathbf{v}_p)_{p \in \mathcal{P}} \in (\mathbb{N}_0^{[1,Q]} \setminus \{j,f\})^{\mathcal{P}} \\ \forall q \neq j, \mathbf{v}(q)+\mathbf{w}(q)=N'_q}}$ is also finite.

$$
\mathbf{w}=(\mathbf{w}_p)_{p \in \mathcal{P}} \in (\mathbb{N}_0^Q)^{\mathcal{P}} \quad \mathbf{v}=(\mathbf{v}_p)_{p \in \mathcal{P}} \in (\mathbb{N}_0^{[1,Q]} \setminus \{j,f\})^{\mathcal{P}} \\
\forall p \in \mathcal{P}^c |\mathbf{w}_p|=k_p \quad \forall q \neq j, f, \mathbf{v}(q)+\mathbf{w}(q)=N'_q
$$

If the coefficients c are rationals, we can replace the derivatives of the auxiliary functions at nonpositive
integers with the formula obtained in Corollary B1:
**Corollary D1.** Let $(\mathbf{N}, \mathbf{N}') \in \mathbb{N}_0^P \times \mathbb{N}_0^Q$, and $(\boldsymbol{\mu}, \boldsymbol{\mu}') \in \mathbb{R}^{+P} \times \mathbb{R}_*^{+Q}$. For $1 \le p \le P$, we write $c_{q,p} = \frac{a_{q,p}}{b_{q,p}} \in \mathbb{Q}$ the irreducible decomposition of $c_{q,p}$ with $a_{q,p} > 0$ and $b_{q,p} > 0$. We set $x_q(\mathbf{c}) = \frac{\text{lcm}(a_{q,p})}{\text{gcd}(b_{q,p})} \in \mathbb{Q}_+^*$, and $\beta_{q,p}(\mathbf{c}) := \frac{x_q(\mathbf{c})}{c_{q,p}} \in \mathbb{N}$ for all $1 \le p \le P$. Then we have

$$
\begin{align*}
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}^c} \in \mathbb{N}_0^{P_c} \\ |\mathbf{k}|=|\mathbf{N}'|+|\mathbf{N}|_{|\mathcal{P}|+|\mathcal{P}|}}} \prod_{p \in \mathcal{P}^c} \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}^c} \in \mathbb{N}_0^{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 x_j(\mathbf{c})^{N'_j} \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( \prod_{\substack{q=1 \\ q \ne j}}^{Q} \left( N'_q \right) \prod_{p=1}^{P} c_{q,p}^{u_{q,p}} \right) \prod_{p=1}^{P} \beta_{j,p}(\mathbf{c})^{\mathbf{R}(\mathbf{N}, j, \mathbf{u})_p} \\
&\qquad \cdot \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_{\substack{\mathbf{p} \in \mathcal{P}^c \\ \mathbf{p} \subseteq [1, P]}} \left( \prod_{p \in \mathcal{P}^c} \mathbf{R}(\mathbf{N}, j, \mathbf{u})_p! \right) \sum_{\substack{\mathbf{k}=(k_p)_{p \in \mathcal{P}^c} \in \mathbb{N}_0^P, k' \in \mathbb{N}_0 \\ |\mathbf{k}|+k'=|\mathbf{R}(\mathbf{N}, j, \mathbf{u})|_{|\mathcal{P}^c|+|\mathcal{P}^c|-1}}} (-1)^{|\mathbf{k}|} \left( \prod_{p \in \mathcal{P}} \frac{\zeta\left(-\mathbf{R}(\mathbf{N}, j, \mathbf{u})_p - k_p, \frac{d_p+v_p}{\beta_{j,p}(\mathbf{c})}\right)}{k_p!} \right) \\
&\qquad \cdot \left( \frac{\zeta'(-N'_j - k', l_j(x_j(\mathbf{c})^{-1}(\mathbf{d}+\mathbf{v})))}{k'!} - \ln(x_j(\mathbf{c})) \frac{\zeta(-N'_j - k', l_j(x_j(\mathbf{c})^{-1}(\mathbf{d}+\mathbf{v})))}{k'!} \right) \\
&\qquad - \sum_{j=1}^{Q} (\gamma - h_{N'_j}) \mu'_j N'_j! \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 \ne j}}^{Q} \left( N'_q \right) \prod_{p=1}^{P} c_{q,p}^{u_{q,p}} \right) \right) \\
&\qquad \cdot \sum_{\substack{\mathcal{P} \subseteq [1, P] \\ \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! \sum_{\substack{\mathbf{k}=(k_p)_{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!},
\end{align*}
$$

where we set

$$
\mathbf{R}(\mathbf{N}, j, \mathbf{u}) := (N_p + \mathbf{u}(p))_{p \in [1, P]},
$$
for all $\mathbf{u} = (\mathbf{u}_q)_{q \in [1,Q] \setminus \{j\}}$, and for all $j \in [1,Q]$,

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

and

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

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

**Remark 1.4.6.** Let $Q = 1$, and $P \ge 1$ a positive integer, we set $l_1(\mathbf{n}) = n_1 + \dots + n_P$ (and thus $c_{1,1} = \dots = c_{1,P} = 1$), $(\boldsymbol{\mu}, \boldsymbol{\mu}') = (\mathbf{1}, \mathbf{1})$, and $(\mathbf{N}, \mathbf{N}') = (N_\Delta, N_\Delta)$ (we recall that $N_\Delta = (N, \dots, N)$). Then we find by the previous corollary that

* For all $\mathcal{P} \subset [1, P]$, and $\mathbf{k} \in \mathbb{N}_0^{\mathcal{P}}$, we have $Q_{\mathbf{N},\mathbf{N}'}^0(1, \mathcal{P}, \mathbf{k}) = 1$ and $Q_{\mathbf{N},\mathbf{N}'}^1(1, \mathcal{P}, \mathbf{k}) = 0$.

* With the same notation as in Corollary D1, we have $\mathbf{x}_1(\mathbf{c}) = 1$, and $\beta_{1,p} = 1$ for all $1 \le p \le P$.

* We have $\mathbf{R}(\mathbf{N}, j, \mathbf{u}) := \mathbf{N}_\Delta \in \mathbb{N}_0^P$.

* In the formula of $Z'(-(N_\Delta, N_\Delta))$ obtained by the previous corollary, we get that the first sum cancels the last sum.
Using the last 4 points, we get that

$$
\begin{align*}
Z'(-(N_{\Delta}, N_{\Delta})) = \sum_{\mathcal{P} \subseteq [1, P]} \frac{N!^{\mathcal{P}|+1}}{|\mathcal{P}|+1} (-1)^{|\mathcal{P}|(N+1)} \sum_{\substack{\mathbf{k}=(k_p)_{p \in \mathcal{P}^c} \in \mathbb{N}_0^{\mathcal{P}^c} \\ |\mathbf{k}|=|\mathcal{P}|(N+1)+|\mathcal{P}|}} \\
\left( \sum_{i \in \mathcal{P}^c} \frac{\zeta'(-N-k_i, d_i)}{k_i!} \prod_{p \in \mathcal{P}^c \setminus \{i\}} \frac{\zeta(-N-k_p, d_p)}{k_p!} \right) \\
+ \sum_{\mathcal{P} \subseteq [1, P]} N!^{\mathcal{P}^c} \sum_{\substack{\mathbf{k}=(k_p)_{p \in \mathcal{P}} \in \mathbb{N}_0^{\mathcal{P}}, k' \in \mathbb{N}_0 \\ |\mathbf{k}|+k'=|\mathcal{P}^c|(N+1)+|\mathcal{P}^c|-1}} (-1)^{|\mathbf{k}|} \frac{\zeta'(-N-k', d'_1)}{k'!} \prod_{p \in \mathcal{P}} \frac{\zeta(-N-k_p, d_p)}{k_p!}.
\end{align*}
$$

This expression matches the one obtained by Onodera in [Ono21, Theorem 4] (with the parameter $d_1 = 0$ in Onodera's theorem).

By considering $(\mathbf{N}, \mathbf{N}') = (\mathbf{0}, \mathbf{0})$ in Theorem D, we get:

**Corollary D2.** Let $(\boldsymbol{\mu}, \boldsymbol{\mu}') \in \mathbb{R}^{+P} \times \mathbb{R}_*^{+Q}$, then

$$
\begin{align*}
Z'(-(\mathbf{0}, \mathbf{0})) = & \\
& \sum_{\mathcal{P} \subseteq [1, P]} \sum_{j=1}^{Q} (-1)^{|\mathcal{P}|} \frac{\mu_j'}{|\boldsymbol{\mu}'|+|\boldsymbol{\mu}|_{\mathcal{P}}} \sum_{\substack{\mathbf{k}=(k_p)_{p \in \mathcal{P}^c} \in \mathbb{N}_0^{\mathcal{P}^c} \\ |\mathbf{k}|=|\mathcal{P}|}} \prod_{p \in \mathcal{P}^c} \frac{\zeta(-k_p, d_p)}{k_p!} \left( \left( \prod_{p \in \mathcal{P}^c} c_{j,p}^{k_p} \right) \gamma(\mu_j' + |\boldsymbol{\mu}|_{\mathcal{P}}) + Q_{\boldsymbol{\mu}, \boldsymbol{\mu}'}^1(j, \mathcal{P}, \mathbf{k}) \right) \\
& + \sum_{\mathcal{P} \subseteq [1, P]} \sum_{j=1}^{Q} (-1)^{|\mathcal{P}|} \frac{\mu_j'}{|\boldsymbol{\mu}'|+|\boldsymbol{\mu}|_{\mathcal{P}}} \sum_{\substack{\mathbf{k}=(k_p)_{p \in \mathcal{P}^c} \in \mathbb{N}_0^{\mathcal{P}^c} \\ |\mathbf{k}|=|\mathcal{P}|}} \prod_{p \in \mathcal{P}^c} c_{j,p}^{k_p} \left( \sum_{i \in \mathcal{P}^c} \mu_i \frac{\zeta'(-k_i, d_i)}{k_i!} \prod_{p \in \mathcal{P}^c \setminus \{i\}} \frac{\zeta(-k_p, d_p)}{k_p!} \right) \\
& + \sum_{j=1}^{Q} \mu_j' (\zeta^B)'(0, d'_j | \mathbf{c}_j) - \sum_{j=1}^{Q} \gamma \mu_j' \sum_{\mathcal{P} \subseteq [1, P]} (-1)^{|\mathcal{P}|} \prod_{p \in \mathcal{P}} c_{j,p}^{-1} \sum_{\substack{\mathbf{k}=(k_p)_{p \in \mathcal{P}^c} \in \mathbb{N}_0^{\mathcal{P}^c} \\ |\mathbf{k}|=|\mathcal{P}|}} \prod_{p \in \mathcal{P}^c} c_{j,p}^{k_p} \frac{\zeta(-k_p, d_p)}{k_p!},
\end{align*}
$$

with

$$
\begin{align*}
Q_{\boldsymbol{\mu}, \boldsymbol{\mu}'}^1(j, \mathcal{P}, \mathbf{k}) = & \sum_{\substack{f=1 \\ f \neq j}}^{Q} \mu_f' \sum_{\substack{\mathbf{w}'=(w'_p)_{p \in \mathcal{P}^c} \in \prod_{p \in \mathcal{P}^c} [0, w_p] \\ \gamma |\boldsymbol{\mu}'|_{[1,Q] \setminus \{j\}} \prod_{p \in \mathcal{P}^c} c_{j,p}^{k_p}}} F_{\mathcal{P},j,f,\mathbf{0},(w'_p \mathbf{e}_f+(k_p-w'_p)\mathbf{e}_j)_{p \in \mathcal{P}^c}(\mathbf{0},\mathbf{0})} \left( \prod_{p \in \mathcal{P}^c} \binom{k_p}{w'_p} c_{f,p}^{w'_p} c_{j,p}^{k_p-w'_p} \right) \\
& + \gamma |\boldsymbol{\mu}'|_{[1,Q] \setminus \{j\}} \prod_{p \in \mathcal{P}^c} c_{j,p}^{k_p},
\end{align*}
$$

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

**Remark 1.4.7.** The constants $F_{\mathcal{P},j,f,\mathbf{0},(w'_p \mathbf{e}_f+(k_p-w'_p)\mathbf{e}_j)_{p \in \mathcal{P}^c}(\mathbf{0},\mathbf{0})}$ in the previous corollary are linked with the partial fraction decomposition of

$$x^{-1+k_p} \prod_{p \in \mathcal{P}} (c_{j,p} + c_{f,p}x)^{-1}.$$

**Remark 1.4.8.** It is worth noting that Matsumoto and Weng obtained in [MW02, Theorem E] a similar expression of the derivative value at $s = 0$ of a zeta function defined by two polynomials in $\mathbb{C}[X]$.

By taking $\boldsymbol{\mu} = \mathbf{0}$, and $\boldsymbol{\mu}' = \mathbf{1}$, we get $Z'(-(\mathbf{0}, \mathbf{0})) = \partial_s(\zeta^{Sh}(s, (\mathbf{c}_1, ..., \mathbf{c}_Q), \mathbf{d}')|_{s=0})$, which corresponds to the first derivative with respect to $s$ of the Shintani zeta function at $s = 0$, defined in 1.2.13. Let's recall
that in Definition 1.2.16, and in Definition 1.2.14, we have defined multigamma function. Therefore, we have $(\partial_s \zeta^B(s, d|c_1, ..., c_P))|_{s=0} = \ln(\Gamma_P(d|c_1, ..., c_P))$, and $\partial_s (\zeta^{Sh}(s, \mathbf{d}'|\mathbf{c}))|_{s=0} = \ln(\Gamma_P(\mathbf{d}'|\mathbf{c}))$. From the previous corollary, and by definition of these multigamma functions, we get the logarithm of the Shintani multigamma function $\Gamma_P^{Sh}(\mathbf{d}'|\mathbf{c}_1, ..., \mathbf{c}_Q)$ has an explicit expression using logarithm of the Barnes zeta function.

**Corollary D3.** We have

$$ \ln(\Gamma_P(\mathbf{d}'|\mathbf{c})|_{s=0} = \sum_{\mathcal{P} \subsetneq [1, P]} \sum_{j=1}^{Q} \frac{(-1)^{|\mathcal{P}|}}{Q} \sum_{\mathbf{k}=(k_p)_{p \in \mathcal{P}^c} \in \mathbb{N}_0^{\mathcal{P}^c} \atop |\mathbf{k}|=|\mathcal{P}|} \prod_{p \in \mathcal{P}^c} \frac{\zeta(-k_p, d_p)}{k_p!} \left( \gamma \left( \prod_{p \in \mathcal{P}^c} c_{j,p}^{k_p} \right) + Q^1(j, \mathcal{P}, \mathbf{k}) \right) + \sum_{j=1}^{Q} \ln(\Gamma_P(d'_j|\mathbf{c}_j)) - \gamma \sum_{j=1}^{Q} \sum_{\mathcal{P} \subsetneq [1, P]} (-1)^{|\mathcal{P}|} \prod_{p \in \mathcal{P}} c_{j,p}^{-1} \sum_{\mathbf{k}=(k_p)_{p \in \mathcal{P}^c} \in \mathbb{N}_0^{\mathcal{P}^c} \atop |\mathbf{k}|=|\mathcal{P}|} \prod_{p \in \mathcal{P}^c} c_{j,p}^{k_p} \frac{\zeta(-k_p, d_p)}{k_p!}, $$

with

$$ Q^1(j, \mathcal{P}, \mathbf{k}) = \sum_{\substack{f=1 \\ f \neq j}}^{Q} \sum_{\mathbf{w}'=(w'_p)_{p \in \mathcal{P}^c} \in \prod_{p \in \mathcal{P}^c} [0, k_p]} F_{\mathcal{P}, j, f, \mathbf{0}, (w'_p e_f + (k_p - w'_p) e_j)_{p \in \mathcal{P}^c}} (\mathbf{0}, \mathbf{0}) \left( \prod_{p \in \mathcal{P}^c} \binom{k_p}{w'_p} c_{f, p}^{w'_p} c_{j, p}^{k_p - w'_p} \right) + \gamma(Q-1) \prod_{p \in \mathcal{P}^c} c_{j, p}^{k_p}. $$

Applying Corollary A1 and Corollary D2 to the two Witten zeta functions $\zeta_{\mathfrak{g}_2}$ and $\zeta_{\mathfrak{so}(5)}(s)$, we get the following formulas:

**Theorem E.** We have the following formulas for the special values of $\zeta_{\mathfrak{g}_2}(s)$ and $\zeta_{\mathfrak{so}(5)}(s)$ at $s = 0$:

$$ \zeta_{\mathfrak{g}_2}(0) = \frac{29}{60}, \\ \zeta_{\mathfrak{so}(5)}(0) = \frac{7}{18}. $$

Moreover, we have the following formulas for the special values of $\zeta'_{\mathfrak{g}_2}(s)$ and $\zeta'_{\mathfrak{so}(5)}(s)$ at $s = 0$:

$$ \zeta'_{\mathfrak{g}_2}(0) = \frac{29}{60} \ln(120) + 2\gamma\zeta(0)^2 + \frac{1}{4}\zeta(0)\zeta'(0) - 4\gamma\zeta(-1) + \frac{1}{5}\zeta(-1)(\ln(3) + 2\ln(2)) - \frac{14}{5}\zeta(-1) \\ - \frac{11}{9}\ln(2) - \frac{23}{24}\ln(3) - \frac{1}{2}\ln(\pi) - 5\zeta'(0) + 3\zeta'(-1) - \frac{4}{3}\zeta'(0, \frac{5}{3}) - \frac{1}{3}\zeta'(0, \frac{4}{3}) \\ + \zeta'(-1, \frac{5}{3}) + \zeta'(-1, \frac{4}{3}) - \frac{1}{6}\zeta'(0, \frac{7}{6}) + \frac{1}{6}\zeta'(0, \frac{5}{6}), $$

$$ \zeta'_{\mathfrak{so}(5)}(0) = \frac{7}{18} \ln(6) - \frac{11}{24} \ln(2) - \frac{1}{4} \ln(\pi) + \left( \frac{1}{2} \zeta(0)^2 - \frac{1}{2} \zeta(-1) \right) \gamma - \frac{11}{4} \zeta'(0) - \frac{13}{6} \zeta'(-1). $$

We can simplify the formula of $\zeta'_{\mathfrak{so}(5)}(0)$:

$$ \zeta'_{\mathfrak{so}(5)}(0) = -\frac{13}{6} \left( \frac{1}{12} - \ln(A) \right) + \frac{1}{6}\gamma + \frac{47}{36}\ln(2) + \frac{7}{18}\ln(3) + \frac{9}{8}\ln(\pi) \approx 2.94635... $$

with $A$ being the Gleicher-Kinkelin constant [OEI23b], and $\gamma$ being Euler's constant [OEI23a].

Using last formula, and by taking the expression of the constant $C$ in Theorem 1.2.35, we get:

$$ C = \frac{e^{\frac{1}{6}\gamma} 2^{\frac{35}{36}} \pi^{\frac{9}{8}} A^{\frac{13}{6}} \Gamma\left(\frac{1}{4}\right)^{\frac{1}{6}} \zeta\left(\frac{3}{2}\right)^{\frac{1}{12}}}{3^{\frac{5}{72}} e^{\frac{13}{72}} \sqrt{\pi}}. $$
We now obtain an explicit version of the Theorem 1.2.35 proved by Bridges, Brindle, Bringmann and Franke in [BBBF23, Theorem 1.3]:

**Corollary F.** For all positive integers $N \ge 1$, we have

$$r_{\mathfrak{so}(5)}(n) = \lim_{n \to +\infty} \frac{C}{n^{\frac{7}{12}}} \exp \left( A_1 n^{\frac{1}{3}} + A_2 n^{\frac{2}{9}} + A_3 n^{\frac{1}{9}} + A_4 \right) \left( 1 + \sum_{j=2}^{N+1} \frac{B_j}{n^{\frac{j-1}{9}}} + O_N \left( n^{-\frac{N+1}{9}} \right) \right),$$

with

$$b = \frac{7}{12}, \quad C = \frac{2^{\frac{35}{36}} \pi^{\frac{9}{8}} A^{\frac{13}{6}} \Gamma(\frac{1}{4})^{\frac{1}{6}} \zeta(\frac{3}{2})^{\frac{1}{12}}}{3^{\frac{5}{2}} e^{\frac{13}{24} + \frac{1}{9}\gamma} \sqrt{\pi}}, \quad A_1 = \frac{3^{\frac{4}{3}} \Gamma(\frac{1}{4})^{\frac{4}{3}} \zeta(\frac{3}{2})^{\frac{2}{3}}}{2^{\frac{8}{3}}},$$

$$A_2 = \frac{2^{\frac{8}{9}} (2^{\frac{1}{3}} + 1) \Gamma(\frac{1}{3}) \zeta(\frac{1}{3}) \zeta(\frac{4}{3})}{3^{\frac{7}{9}} \Gamma(\frac{1}{4})^{\frac{4}{9}} \zeta(\frac{3}{2})^{\frac{2}{9}}}, \quad A_3 = \frac{2^{\frac{40}{9}} (2^{\frac{1}{3}} + 1)^2 \Gamma(\frac{1}{3})^2 \zeta(\frac{1}{3})^2 \zeta(\frac{4}{3})^2}{3^{\frac{44}{9}} \Gamma(\frac{1}{4})^{\frac{20}{9}} \zeta(\frac{3}{2})^{\frac{10}{9}}},$$

$$A_4 = \frac{2^8 (2^{\frac{1}{3}} + 1)^3 \Gamma(\frac{1}{3})^3 \zeta(\frac{1}{3})^3 \zeta(\frac{4}{3})^3}{3^8 \Gamma(\frac{1}{4})^4 \zeta(\frac{3}{2})^2}.$$

We now wish to apply the results obtained in Theorem E to derive an asymptotic formula for $r_{\mathfrak{g}_2}(n)$, which we get through Theorem 1.2.33 obtained by Bridges, Brindle, Bringmann, and Franke in [BBBF23, Theorem 4.4]. We first set

$$\begin{aligned} l_1^*(x_1, x_2) &= x_1 + x_2 + x_3 + 2x_4, \\ l_2^*(x_1, x_2) &= x_1 + 2x_2 + 3x_3 + 4x_4, \end{aligned}$$

where we set

$$\begin{aligned} \omega_{\beta} := &120^{\frac{1}{3}} \frac{\Gamma(\frac{4}{5})}{\Gamma(\frac{1}{5})^4} \zeta(\frac{1}{5}) \left( \int_{[0,1]^3} (x_2 x_3 x_4)^{-\frac{4}{5}} (l_1^*(1, x_2, x_3, x_4)^{-\frac{4}{5}} + l_2^*(1, x_2, x_3, x_4)^{-\frac{4}{5}}) dx_2 dx_3 dx_4 \right. \\ &+ \int_{[0,1]^3} (x_1 x_3 x_4)^{-\frac{4}{5}} (l_1^*(x_1, 1, x_3, x_4)^{-\frac{4}{5}} + l_2^*(x_1, 1, x_3, x_4)^{-\frac{4}{5}}) dx_1 dx_3 dx_4 \\ &+ \int_{[0,1]^3} (x_1 x_2 x_4)^{-\frac{4}{5}} (l_1^*(x_1, x_2, 1, x_4)^{-\frac{4}{5}} + l_2^*(x_1, x_2, 1, x_4)^{-\frac{4}{5}}) dx_1 dx_2 dx_4 \\ &\left. + \int_{[0,1]^3} (x_1 x_2 x_3)^{-\frac{4}{5}} (l_1^*(x_1, x_2, x_3, 1)^{-\frac{4}{5}} + l_2^*(x_1, x_2, x_3, 1)^{-\frac{4}{5}}) dx_1 dx_2 dx_3 \right), \end{aligned}$$

and

$$\begin{aligned} \omega_{\alpha} = &120^{\frac{1}{3}} \frac{\Gamma(\frac{2}{3})^2}{\Gamma(\frac{1}{3})} \left( \int_{[0,1]^3} (x_1 x_2 x_3)^{-\frac{2}{3}} \prod_{p=1}^{2} l_p^*(x_1, x_2, x_3, 1)^{-\frac{2}{3}} dx_1 dx_2 dx_3, \right. \\ &+ \int_{[0,1]^3} (x_1 x_2 x_4)^{-\frac{2}{3}} \prod_{p=1}^{2} l_p^*(x_1, x_2, 1, x_4)^{-\frac{2}{3}} dx_1 dx_2 dx_4 \\ &+ \int_{[0,1]^3} (x_1 x_3 x_4)^{-\frac{2}{3}} \prod_{p=1}^{2} l_p^*(x_1, 1, x_3, x_4)^{-\frac{2}{3}} dx_1 dx_3 dx_4 \\ &\left. + \int_{[0,1]^3} (x_2 x_3 x_4)^{-\frac{2}{3}} \prod_{p=1}^{2} l_p^*(1, x_2, x_3, x_4)^{-\frac{2}{3}} dx_2 dx_3 dx_4 \right). \end{aligned}$$

We then state an asymptotic formula for the number of representation $r_{\mathfrak{g}_2}(n)$ of the exceptional algebra $\mathfrak{g}_2$, some of the coefficients of the formula have an expression using the residues $\omega_{\alpha}$ and $\omega_{\beta}$, and the special values $\zeta_{\mathfrak{g}_2}(0)$ and $\zeta'_{\mathfrak{g}_2}(0)$.
**Theorem G.** Let $N \in \mathbb{N}$. When $n$ tends to $+\infty$, we have

$$r_{g_2}(n) = \frac{C}{n^{b}} \exp \left( A_1 n^{\frac{1}{4}} + A_2 n^{\frac{3}{20}} + A_3 n^{\frac{1}{20}} \right) \left( 1 + \sum_{j=2}^{N+1} \frac{B_j}{n^{\frac{j-1}{20}}} + O \left( n^{-\frac{N+1}{20}} \right) \right),$$

with $B_j$ being explicit constants, and $C, b, A_1, A_2, A_3$ such that:

$$C = \frac{e^{\zeta'_{g_2}(0)} \left(\omega_{\alpha} \Gamma\left(\frac{4}{3}\right) \zeta\left(\frac{4}{3}\right)\right)^{\frac{1-6\zeta_{g_2}(0)}{8}} \sqrt{3}}{\sqrt{8\pi}}, \quad b = \frac{6-6\zeta_{g_2}(0)+1}{8},$$

$$K_2 = \frac{3}{4} \cdot \frac{\omega_{\beta} \Gamma\left(\frac{6}{5}\right) \zeta\left(\frac{6}{5}\right)}{\left(\omega_{\alpha} \Gamma\left(\frac{4}{3}\right) \zeta\left(\frac{4}{3}\right)\right)^{\frac{3}{20}}}, \quad K_3 = -\frac{3}{160} \cdot \frac{\left(\omega_{\beta} \Gamma\left(\frac{6}{5}\right) \zeta\left(\frac{6}{5}\right)\right)^2}{\left(\omega_{\alpha} \Gamma\left(\frac{4}{3}\right) \zeta\left(\frac{4}{3}\right)\right)^{\frac{21}{20}}},$$

$$A_1 := 4 \left( \omega_{\alpha} \Gamma \left( \frac{4}{3} \right) \zeta \left( \frac{4}{3} \right) \right)^{\frac{3}{4}}, \quad A_2 := \frac{\omega_{\beta} \Gamma \left( \frac{1}{5} \right) \zeta \left( \frac{6}{5} \right)}{\left( \omega_{\alpha} \Gamma \left( \frac{1}{3} \right) \zeta \left( \frac{4}{3} \right) \right)^{\frac{3}{20}}},$$

$$A_3 := K_3 + 3 \left( \binom{-1/3}{1} K_3 + \binom{-1/3}{2} \frac{K_2^2}{\left(\omega_{\alpha} \Gamma\left(\frac{4}{3}\right) \zeta\left(\frac{4}{3}\right)\right)^{\frac{3}{4}}} \right) + \frac{5 \left(\omega_{\beta} \Gamma\left(\frac{6}{5}\right) \zeta\left(\frac{6}{5}\right)\right)}{\left(\omega_{\alpha} \Gamma\left(\frac{4}{3}\right) \zeta\left(\frac{4}{3}\right)\right)^{\frac{18}{20}}} \binom{-1/5}{1} K_2$$

$$\zeta'_{g_2}(0) = \frac{29}{60} \ln(120) + 2\gamma\zeta(0)^2 + \frac{1}{4}\zeta(0)\zeta'(0) - 4\gamma\zeta(-1) + \frac{1}{5}\zeta(-1)(\ln(3) + 2\ln(2)) - \frac{14}{5}\zeta'(-1) \\ - \frac{11}{9}\ln(2) - \frac{23}{24}\ln(3) - \frac{1}{2}\ln(\pi) - 5\zeta'(0) + 3\zeta'(-1) - \frac{4}{3}\zeta'\left(0, \frac{5}{3}\right) - \frac{1}{3}\zeta'\left(0, \frac{4}{3}\right) \\ + \zeta'\left(-1, \frac{5}{3}\right) + \zeta'\left(-1, \frac{4}{3}\right) - \frac{1}{6}\zeta'\left(0, \frac{7}{6}\right) + \frac{1}{6}\zeta'\left(0, \frac{5}{6}\right),$$

$$\zeta_{g_2}(0) = \frac{29}{60}.$$

**Remark 1.4.9.** The $B_j$ can be computed through the coefficients $b_j$ obtained by [BBBF23, Lemma 3.6].
<figure><img src="image_15.png" /></figure>
# Chapter 2

## Crandall's Expansion of $Z(s, s')$

We shall fix in this chapter coefficients $c = (c_{q,p})_{q \in \llbracket 1,Q \rrbracket, p \in \llbracket 1,P \rrbracket}$, $\mathbf{d} = (d_p)_{p \in \llbracket 1,p \rrbracket}$, $\mathbf{d}' = (d'_q)_{q \in \llbracket 1,Q \rrbracket}$ such that $d_p \in H_0$ for all $1 \le p \le P$, and we assume that for all $1 \le q \le Q$,

$$ \forall q \in \llbracket 1, Q \rrbracket, \quad d'_q = c_q \cdot \mathbf{d} = \sum_{p=1}^{P} c_{q,p} d_p. $$

Moreover, we will consider a $P+Q$-tuple integer $(\mathbf{N}, \mathbf{N}') \in \mathbb{N}_0^P \times \mathbb{N}_0^Q$, and a direction $(\boldsymbol{\mu}, \boldsymbol{\mu}') \in \mathbb{R}^+ \times \mathbb{R}_*^+. $ All of the bounds that we will state through this chapter will depend on the above constants. However, we will not note that dependence for the sake of clarity.

We will first study the meromorphic extension of $Z(s, s')$ and examine its singularities in the proof of Proposition 1.3.9. Next, we will decompose the meromorphic function $Z(s, s')$ using a Crandall expansion, and we will obtain an extension formula for the univariate function $Z(-\mathbf{N} + \boldsymbol{\mu}s, -\mathbf{N} + \boldsymbol{\mu}'s)$. More specifically, we will obtain a decomposition of this meromorphic function as a sum between a holomorphic function that vanishes at $s = 0$, and a meromorphic function corresponding to an Erdélyi-type series. This extension formula will be crucial in the proofs of Theorems A and D.

### 2.1 Proof of Proposition 1.3.9

The meromorphic extension of $Z(s, s')$ and the form of the singularities follow from Theorem 1.2.25. We aim to show here that $Z(s, s')$ is absolutely convergent on the domain $\mathcal{D}$ introduced in Proposition 2.1.1. It is enough to show that the series describing $Z(s, s')$ is normally convergent on every compact subset of $\mathcal{D}$. We intend to prove the following proposition, which is more general than what we aim to prove in Proposition 1.3.9:

**Proposition 2.1.1.** Let $P, Q \ge 1$ be two positive integers, and $\gamma = (\gamma_q)_{q \in \llbracket 1,Q \rrbracket} \in (H_0^P)^Q$, $\alpha = (\alpha_p) \in H_0^P$, $\beta = (\beta_q) \in H_0^Q$. We set

$$ \mathcal{D} = \mathcal{D}_{P,Q} := \left\{ (\mathbf{s}, \mathbf{s}') \in \mathbb{C}^P \times \mathbb{C}^Q \middle| \sum_{p \in \mathcal{P}} \sigma_p + \sigma'_1 + \dots + \sigma'_Q > |\mathcal{P}|, \emptyset \ne \mathcal{P} \subset \llbracket 1, P \rrbracket \right\}. $$

Then the function $Z_{\gamma, \alpha, \beta}^{P,Q}(\mathbf{s}, \mathbf{s}') := \sum_{\mathbf{n} \in \mathbb{N}_0^P} \prod_{p=1}^P (n_p + \alpha_p)^{-s_p} \prod_{q=1}^Q \left( \sum_{p=1}^P \gamma_{q,p} n_p + \beta_q \right)^{-s'_q}$ normally converges on all compact sets of $\mathcal{D}$.

*Proof.* All the constants involved in the proof of this proposition will implicitly depend on the choice of the coefficients $(\gamma_q)_{q \in \llbracket 1,Q \rrbracket}$, $\alpha$, and $\beta$. We will not show this dependency to improve the clarity of the proof. We will prove this result by induction. The case $P = 1$ is trivial.
Let's assume that the result is true for all $k < P$. Let $(\mathbf{s}, \mathbf{s}') \in H_0^{P+Q}$, and $\mathbf{n} \in \mathbb{N}_0^P$. We then have the following upper bound

$$ (n_p + 1) \ll n_p + \mathrm{Re}(\alpha_p) \le |n_p + \alpha_p| \le n_p + |\alpha_p| \ll (n_p + 1). $$

Thus, we have that

$$ \sum_{p=1}^{P} n_p + P \ll \sum_{p=1}^{P} \mathrm{Re}(\gamma_{q,p})n_p + \mathrm{Re}(\beta_q) \le \left| \sum_{p=1}^{P} \gamma_{q,p}n_p + \beta_q \right| \le \sum_{p=1}^{P} |\gamma_{q,p}|n_p + |\beta_q| \ll \sum_{p=1}^{P} n_p + P. $$

Taking the product over $[1, P]$ and $[1, Q]$, we find that

$$ \left| \prod_{p=1}^{P} (n_p + d_p)^{-s_p} \prod_{q=1}^{Q} (l_q(\mathbf{n}) + d'_q)^{-s'_q} \right| \ll e^{(\sum_{p=1}^{P} |\tau_p| + \sum_{q=1}^{Q} |\tau'_q|)\pi/2} \prod_{p=1}^{P} (n_p + 1)^{-\sigma_p} (P + \sum_{p=1}^{P} n_p)^{-(\sigma'_1 + \dots + \sigma'_Q)}. $$

We now aim to show that the series $\sum_{\mathbf{n} \in \mathbb{N}_0^P} \prod_{p=1}^P (n_p + 1)^{-\sigma_p} (P + \sum_{p=1}^P n_p)^{-(\sigma'_1 + \dots + \sigma'_Q)}$ is normally convergent on every compact sets of $\mathcal{D}$. This series can be split into a finite sum of series of the form

$$ \sum_{0 \le n_{f(1)} \le \dots \le n_{f(P)}} \prod_{p=1}^{P} (n_p + 1)^{-\sigma_p} (P + \sum_{p=1}^{P} n_p)^{-(\sigma'_1 + \dots + \sigma'_Q)}, $$

with $f \in \mathfrak{S}_P$, and in terms of terms of the form $Z_{1,1,P}^{k,1}$, $(\sigma_1 + \dots + \sigma_{i_1}, \dots, \sigma_{i_{k-1}+1} + \dots + \sigma_{i_k}), \sigma'_1 + \dots + \sigma'_Q)$ with $1 = i_1 < i_2 < \dots < i_k = P$ and $(i_1, \dots, i_k) \ne (1, \dots, P)$ (which in particular implies $k < P$). By induction principle, the Dirichlet series $Z_{1,1,P}^{k,1}$, $(\sigma_1 + \dots + \sigma_{i_1}, \dots, \sigma_{i_{k-1}+1} + \dots + \sigma_{i_k}), \sigma'_1 + \dots + \sigma'_Q)$ converges normally on every compact sets of

$$ \left\{ (\mathbf{s}, \mathbf{s}') \in \mathbb{C}^P \times \mathbb{C}^Q \left| \sum_{p \in \mathcal{P}} (\sigma_{i_p} + \dots + \sigma_{i_{p+1}-1}) \right| + \sigma'_1 + \dots + \sigma'_Q > |\mathcal{P}|, \mathcal{P} \subset \{i_1, \dots, i_{k-1}\} \right\}, $$

and we see that the above set do contains the set $\mathcal{D}_{P,Q}$.

Let's consider the trivial permutation $f = \mathrm{Id} \in \mathfrak{S}_P$. By a change of variables we see that

$$ \sum_{0 \le n_{f(1)} \le \dots \le n_{f(P)}} \prod_{p=1}^{P} (n_p + 1)^{-\sigma_p} (P + \sum_{p=1}^{P} n_p)^{-(\sigma'_1 + \dots + \sigma'_Q)} = \sum_{\mathbf{n} \in \mathbb{N}_0^P} \prod_{p=1}^{P} \left(1 + \sum_{i=1}^{p} n_i\right)^{-\sigma_p} \left(P + \sum_{p=1}^{P} p n_p\right)^{-(\sigma'_1 + \dots + \sigma'_Q)}. $$

By a previous argument, we get that:

$$ \sum_{\mathbf{n} \in \mathbb{N}_0^P} \prod_{p=1}^{P} \left(1 + \sum_{i=1}^{p} n_i\right)^{-\sigma_p} \left(P + \sum_{p=1}^{P} p n_p\right)^{-(\sigma'_1 + \dots + \sigma'_Q)} \ll \zeta_P^{EZ}(\sigma_1, \dots, \sigma_{P-1}, \sigma'_1 + \dots + \sigma'_Q + \sigma_P). $$

By Lemma 1.2.23, we find that the right hand term is normally convergent on all compact sets of

$$ \{(\mathbf{s}, \mathbf{s}') \in \mathbb{C}^P \times \mathbb{C}^Q | \sigma_p + \dots + \sigma_P + \sigma'_1 + \dots + \sigma'_Q > P + 1 - p, 1 \le p \le P\}. $$

We then get that the series $\sum_{0 \le n_{f(1)} \le \dots \le n_{f(P)}} \prod_{p=1}^{P} (n_p + 1)^{-\sigma_p} (P + \sum_{p=1}^{P} n_p)^{-(\sigma'_1 + \dots + \sigma'_Q)}$, with $f \in \mathfrak{S}_P$ converges normally on all compact subset of

$$ \{(\mathbf{s}, \mathbf{s}') \in \mathbb{C}^P \times \mathbb{C}^Q | \sigma_{f(p)} + \dots + \sigma_{f(P)} + \sigma'_1 + \dots + \sigma'_Q > P + 1 - p, 1 \le p \le P\}. $$

We then get that $Z_{\gamma, \alpha, \beta}^{P,Q}$ converges normally on all compact sets of

$$ \left\{ (\mathbf{s}, \mathbf{s}') \in \mathbb{C}^P \times \mathbb{C}^Q \left| \sum_{p \in \mathcal{P}} \sigma_p + \sigma'_1 + \dots + \sigma'_Q > |\mathcal{P}|, \emptyset \ne \mathcal{P} \subset [1, P] \right. \right\}. $$

□
To obtain the meromorphic extension of the multizeta function $Z(s, s')$, and the singularities of this multizeta function, we can simply use Theorem 1.2.25. Indeed, via Remark 1.3.8, the multizeta function $Z(s, s')$ is of the form $\zeta^{HL}(0, \tilde{d}, \tilde{c}, (s_1, ..., s_P, s'_1, ..., s'_Q))$ as introduced in Definition 1.2.24. Therefore, we can apply Theorem 1.2.25 to this multizeta function.

## 2.2 Statement of the extension formula for $Z_{\mathbf{N},\mathbf{N}'}(s)_{\mu,\mu'}$

We aim to establish an extension formula for $Z_{\mathbf{N},\mathbf{N}'}(s)_{\mu,\mu'}$. Before expressing this formula, we need to introduce the functions $h_{\mathbf{N},\mathbf{N}',\mathcal{P},j,\mathbf{k}}(s)_{\mu,\mu'}$, which will appear in the extension formula.

**Definition 2.2.1.** Let $\mathcal{P} \subsetneq [1, P]$, $j \in [1, Q]$, and $\mathbf{k} = (k_p)_{p \in \mathcal{P}} \in \mathbb{N}_0^{\mathcal{P}^c}$. We set $l_p^*(\mathbf{x}) = c_{1,p}x_1 + \dots + c_{Q,p}x_Q$ for all $1 \le p \le P$. For all $s = \sigma + i\tau \in \mathbb{C}$, with $\sigma \gg 1$, we consider

$$h_{\mathbf{N},\mathbf{N}',\mathcal{P},j,\mathbf{k}}(s) = \frac{1}{\prod_{q=1}^{Q} \Gamma(-N'_q + \mu'_q s)} \cdot \int_0^1 \dots \int_0^1 \prod_{q=1}^{Q} x_q^{-N'_q + \mu'_q s - 1} \prod_{p \in \mathcal{P}} l_p^*(\hat{\mathbf{x}}^j)^{-N_p + \mu_p s - 1} \prod_{p \in \mathcal{P}^c} l_p^*(\hat{\mathbf{x}}^j)^{k_p} dx_1 \dots \widehat{dx_j} \dots dx_Q,$$

and

$$h_{\mathbf{N},\mathbf{N}',[1,P],j}(s) = \frac{1}{\prod_{q=1}^{Q} \Gamma(-N'_q + \mu'_q s)} \int_0^1 \dots \int_0^1 \prod_{q=1}^{Q} x_q^{-N'_q + \mu'_q s - 1} \prod_{p=1}^{P} l_p^*(\hat{\mathbf{x}}^j)^{-N_p + \mu_p s - 1} dx_1 \dots \widehat{dx_j} \dots dx_Q,$$

where we have set $dx_1 \dots \widehat{dx_j} \dots dx_Q = dx_1 \dots dx_{j-1} dx_{j+1} \dots dx_Q$, and $\hat{\mathbf{x}}^j = (x_1, \dots, x_{j-1}, 1, x_{j+1}, \dots, x_Q)$, with $1 \le j \le Q$.

We now state in the following proposition some results about this function. Particularly, we shall state an upper bound of these functions in order to control later a series, and we shall give an expression of these functions in order to compute the values at $s = 0$ of $h_{\mathbf{N},\mathbf{N}',\mathcal{P},j,\mathbf{k}}(s)$ and of $h'_{\mathbf{N},\mathbf{N}',\mathcal{P},j,\mathbf{k}}(s)$.

**Proposition 2.2.2.** 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 $\mathcal{P} \subsetneq [1, P]$, $j \in [1, Q]$ and $\mathbf{k} \in \mathbb{N}_0^{\mathcal{P}^c}$. We have that

i) $h_{\mathbf{N},\mathbf{N}',\mathcal{P},j,\mathbf{k}}$ and $h_{\mathbf{N},\mathbf{N}',[1,P],j}$ are holomorphic on $\mathbb{C}$.

ii) For all positive integer $r \in \mathbb{N}$, we have the following upper bound uniformly for $s \in D_0(r)$:

$$|h_{\mathbf{N},\mathbf{N}',\mathcal{P},j,\mathbf{k}}(s)| \ll (QM)^{|\mathbf{k}|} \quad (2.1)$$

with $M := \max_{1 \le q \le Q, 1 \le p \le P} (|c_{q,p}|)$.
iii) Let $\epsilon > 0$ small enough and $s \in \mathbb{C}$, we have

$$
\begin{align}
h_{\mathbf{N},\mathbf{N}',\mathcal{P},j,\mathbf{k}}(s) = & \sum_{\mathcal{Q} \subset [1,Q] \setminus \{j\}} \sum_{\mathbf{v}=(\mathbf{v}_p)_{p \in \mathcal{P}} \in (\mathbb{N}_0^Q)^\mathcal{P}} \sum_{\substack{p \in \mathcal{P}^c \\ \forall p \in \mathcal{P}^c |\mathbf{w}_p|=k_p}} \left( \prod_{p \in \mathcal{P}^c} \binom{k_p}{\mathbf{w}_p} \prod_{q \in [1,Q]} c_{q,p}^{w_{p,q}} \right) \nonumber \\
& \cdot \left( \prod_{p \in \mathcal{P}} \binom{-N_p + \mu_p s - 1}{|\mathbf{v}_p|} \binom{|\mathbf{v}_p|}{\mathbf{v}_p} \prod_{q \in \mathcal{Q}} c_{q,p}^{v_{p,q}} \right) \frac{1}{\prod_{q \in [1,Q] \setminus \{j\}} \Gamma(-N'_q + \mu'_q s) \prod_{q \in \mathcal{Q}} (-N'_q + \mu'_q s + \mathbf{v}(q) + \mathbf{w}(q))} \nonumber \\
& \cdot \int_{\epsilon}^{1} \dots \int_{\epsilon}^{1} \prod_{q \in \mathcal{Q}^c} x_q^{-N'_q + \mu'_q s - 1 + \mathbf{w}(q)} \prod_{p \in \mathcal{P}} \left( c_{j,p} + \sum_{f \in \mathcal{Q}^c} c_{f,p} x_f \right)^{-N_p + \mu_p s - 1 - |\mathbf{v}_p|} \prod_{q \in \mathcal{Q}^c} dx_q, \tag{2.2}
\end{align}
$$

with the notations $\mathbf{v}(q) = \sum_{p \in \mathcal{P}} v_{p,q}$ for all $q \in \mathcal{Q}$, and $\mathbf{w}(q) = \sum_{p \in \mathcal{P}^c} w_{p,q}$ for all $q \in [1, Q]$.
Furthermore, we can differentiate the general term of the previous series with respect to $s$ to obtain
an expression for $\partial_s h_{\mathbf{N},\mathbf{N}',\mathcal{P},j,\mathbf{k}}(s)$.

**Remark 2.2.3.** 1) Point iii) of this proposition will be critical for evaluating $h_{\mathbf{N},\mathbf{N}',\mathcal{P},j,\mathbf{k}}(s)$ and $\partial_s h_{\mathbf{N},\mathbf{N}',\mathcal{P},j,\mathbf{k}}(s)$
at $s=0$ in the sub-sections 3.2.2 and 3.2.3.

2) We can obtain results similar to points ii) and iii) of the previous proposition for functions of the form $h_{\mathbf{N},\mathbf{N}',[1,P],j}(s)$, however we will not need to study this type of function in detail.

We can now state the extension formula for $Z_{\mathbf{N},\mathbf{N}'}(s)$. We will decompose this analytic function into two parts that will depend on $s$ and a sufficiently small free variable $\theta > 0$. The first, denoted as $K_{\mathbf{N},\mathbf{N}'}(\theta, s)$, will be holomorphic in $s$ and will vanish at $s = 0$. The other term will be an Erdélyi-type series (in the style of the series present in the Erdélyi formula, cf. 2.6), denoted as $J_{\mathbf{N},\mathbf{N}'}(\theta, s)$. This second term will contain the hypothetical singularities of $Z_{\mathbf{N},\mathbf{N}'}(s)$, and we will denote the set of these hypothetical singularities as $S(\mathbf{N}, \mathbf{N}')$, the expression of which is

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

**Proposition 2.2.4.** Let $(\mathbf{N}, \mathbf{N}') \in \mathbb{C}^P \times \mathbb{C}^Q$, and $(\boldsymbol{\mu}, \boldsymbol{\mu}') \in \mathbb{R}_+^P \times \mathbb{R}_+^Q$. Then the function $Z_{\mathbf{N}, \mathbf{N}'}(s)$ is meromorphic on $\mathbb{C}$, and his poles are included in $S(\mathbf{N}, \mathbf{N}')$. Furthermore, we have the following analytical continuation formula:

For all $\theta > 0$ small enough, and for all $s \in \mathbb{C} \setminus S(\mathbf{N}, \mathbf{N}')$, we have

$$
Z_{\mathbf{N},\mathbf{N}'}(s) = J_{\mathbf{N},\mathbf{N}'}(\theta, s) + K_{\mathbf{N},\mathbf{N}'}(\theta, s) \quad (2.3)
$$

with $J_{\mathbf{N},\mathbf{N}'}(\theta, s)$ and $K_{\mathbf{N},\mathbf{N}'}(\theta, s)$ such that:
1) For all $s \in \mathbb{C}$, we set

$$
K_{\mathbf{N},\mathbf{N}'}(\theta, s) = \sum_{\emptyset \neq \mathcal{Q} \subset [1, Q]} \sum_{A_{\mathcal{Q}} \subset \mathcal{Q}^c} (-1)^{|\mathcal{Q}^c| - |A_{\mathcal{Q}}|} \prod_{q \in [1, Q] \setminus A_{\mathcal{Q}}} \frac{1}{\Gamma(-N_q' + \mu_q's)} \left( \sum_{\mathbf{n} \in \mathbb{N}_0^P} \prod_{p=1}^P (n_p + d_p)^{N_p - \mu_p s} \cdot \prod_{q \in A_{\mathcal{Q}}} (l_q(\mathbf{n}) + d_q')^{N_q' - \mu_q's} \prod_{q \in [1, Q] \setminus A_{\mathcal{Q}}} \Gamma(-N_q' + \mu_q's, \theta, l_q(\mathbf{n}) + d_q') \right). \quad (2.4)
$$

2) For all $s \in \mathbb{C}\backslash S(\mathbf{N}, \mathbf{N}')$, we set

$$
J_{\mathbf{N},\mathbf{N}'}(\theta, s) = \tag{2.5}
$$

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

Moreover, we have that:

i) $K_{\mathbf{N},\mathbf{N}'}(\theta, s)$ is holomorphic in $s$ on all $\mathbb{C}$, and vanish at $s = 0$.

ii) $J_{\mathbf{N},\mathbf{N}'}(\theta, s)$ is meromorphic in $s$ on all $\mathbb{C}$, with poles inside $S(\mathbf{N}, \mathbf{N}')$. Furthermore, one can compute the derivative with respect to $s$ of $J_{\mathbf{N},\mathbf{N}'}(\theta, s)$ by deriving the series in the expression of $J_{\mathbf{N},\mathbf{N}'}(\theta, s)$ termwise.

**Remark 2.2.5.** 1) $J_{\mathbf{N},\mathbf{N}'}(\theta, s)$ will be the sole contributor when computing the directional value $Z(-(\mathbf{N}, \mathbf{N}'))$, since the function $K_{\mathbf{N},\mathbf{N}'}(\theta, s)$ vanishes at $s = 0$. However, when studying the value of the derivative $Z'(-(\mathbf{N}, \mathbf{N}'))$, we will have to consider the derivative of the term $K_{\mathbf{N},\mathbf{N}'}(\theta, s)$.

2) We will see in the proof of Proposition 2.2.4 that the series

$$
\sum_{\mathbf{n} \in \mathbb{N}_0^P} \prod_{p=1}^P (n_p + d_p)^{N_p - \mu_p s} \prod_{q \in A_Q} (l_q(\mathbf{n}) + d_q')^{N_q' - \mu_q s} \prod_{q \in [1, Q] \setminus A_Q} \Gamma(-N_q' + \mu_q' s, \theta, l_q(\mathbf{n}) + d_q'),
$$

describes a holomorphic function in $s$ on all $\mathbb{C}$ (the series is actually absolutely convergent on all $\mathbb{C}$). In particular, we can consider the derivative of this series with respect to $s$, and evaluate it at $s = 0$ in the proof of Theorem D. Note that we will never attempt to directly compute the values of this series, or the values of its derivative with respect to $s$.

With the help of this proposition, we will then establish that the function $Z_{\mathbf{N},\mathbf{N}'}$ is regular at 0, and will thus have proven by another method than Komori in [KMT10a] the existence of directional values for the class of zeta functions that interest us. We also retrieve some information on the poles of $Z_{\mathbf{N},\mathbf{N}'}$:
**Corollary 2.2.6.** 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 the function $Z_{\mathbf{N}, \mathbf{N}'}$ has poles, which are all simple, of the form

$$s = \frac{n}{\mu_p} \qquad (1 \le p \le P, n \in [1, P])$$
$$s = \frac{n}{|\boldsymbol{\mu}'| + |\boldsymbol{\mu}|_P} \qquad (\emptyset \ne \mathcal{P} \subset [1, P], n \in \mathbb{N}, |\boldsymbol{\mu}'| + |\boldsymbol{\mu}|_P \notin \mathbb{N})$$

To prove this proposition, we will multiply the Dirichlet series $Z(s, s')$ by $\Gamma(s'_1) \dots \Gamma(s'_Q)$. In a way, this is equivalent to consider a multivariable Mellin transform. This strategy is classic for extending and studying the values of the Riemann zeta function [Hid93, Chap. 2], for obtaining an integral representation of a Shintani zeta function in [Hid93, Chap. 2], and for calculating the values of the multiple Hurwitz-type Mordell-Tornheim zeta function [Ono21].

Next, we will proceed to partition the integration domain by introducing a free variable $\theta$. By doing so, we will isolate a neighborhood of the origin from the rest of the integration domain. Finally, we will note that the integral around the neighborhood of the origin can be re-expressed using an Erdélyi formula (see 2.6), and we will also prove that the remaining integrals actually define a holomorphic function in $(\mathbf{s}, \mathbf{s}')$. To partition the integrals appearing in our calculations, we will use the incomplete gamma functions $\Gamma(\mathbf{s}, \theta, \nu)$ and $\gamma(\mathbf{s}, \theta, \nu)$ defined in the Definition 1.3.1.

## 2.3 Directional Crandall's expansion for $Z(s, s')$

### 2.3.1 Lemmata

We shall start by stating a few lemma. Firstly, let's state a classical identity obtained by a multivariable Taylor's theorem:

**Lemma 2.3.1.** Let $X_1, \dots, X_d$ be complex numbers such that $|X_1 + \dots + X_d| < 1$. We then have that

$$(1 + X_1 + \dots + X_d)^{-s} = \sum_{\mathbf{k}=(k_p)_{p \in [1, d]}} \binom{-s}{|\mathbf{k}|} \binom{|\mathbf{k}|}{\mathbf{k}} X_1^{k_1} \dots X_d^{k_d}.$$

The expressions in this chapter will involve a series whose general term will contain $\zeta(s-k, d)$ where $k \in \mathbb{N}_0$, and $d \in H_0$. In [Ono21], Onodera also had these terms in one of his expressions, and he controlled them using the following lemma, in the case where $d \in \mathbb{R}_+^*$:

**Lemma 2.3.2.** [Ono21, Lemma 2.1] Let $\alpha > 0$. We set for all $\delta \in \mathbb{R}_+^*$ and all $r \in \mathbb{N}$,

$$D(\delta, r) := \left\{ s \in \mathbb{C} \mid \min_{n \in \mathbb{N}} |s - n| \ge \delta, |s| \le r \right\}.$$

Then for all $s \in D(\delta, r)$, $k \in \mathbb{N}_0$, $x > 0$, $m \in \mathbb{N}_0$, we have

$$|\partial_s^m \zeta(s-k, d)| \ll k!(k+1)^{r+\alpha}.$$

However, this lemma is not sufficient here, as we want to bound $\zeta(s-k, d)$ with $d \in H_0$. The proof of the previous lemma uses a functional equation satisfied by the Hurwitz zeta function $\zeta(s, d)$ for $d \in [0, 1]$. We cannot prove a similar bound for $d \in H_0$ in such a fashion. We will therefore use Euler-Maclaurin's formula on the Hurwitz zeta function to obtain an inequality similar to the previous lemma.

**Lemma 2.3.3.** Let $\delta \in \mathbb{R}_+^*$, $r \in \mathbb{N}$, and $r' \in [1, +\infty[$. Then we have that

$$\forall s \in D(\delta, r), \forall d \in \overline{H}_{1/r'} \cap D_0(r'), \forall k \in \mathbb{N}_0, \quad |\zeta(s-k, d)| \ll_{r, r', \delta} \left(\frac{e}{\pi}\right)^k k!(k+1)^{r+1}(1+r'^{2k}).$$
*Proof.* Let $r \in \mathbb{N}$, we set $n = r + 1$. We aim to bound $|\zeta(s - k, d)|$ uniformly in $s \in D(\delta, r)$ and in $d \in \bar{H}_{1/r'} \cap D_0(r')$. Let $s \in D(\delta, r)$ and $d \in H_{1/r'} \cap D_0(r')$. We note that $n > -\sigma$. Let $k \in \mathbb{N}_0$, as we have $n + k > -\sigma + k$, we can use the expression for the Hurwitz zeta function from Maclaurin's formula 1.2.6 up to order $n + k$, and evaluate it at $s - k$.

$$ \zeta(s-k, d) = \frac{d^{1-s+k}}{1-s+k} + \sum_{i=0}^{n+k} \binom{-s+k}{i} \frac{(-1)^i B_{i+1}}{i+1} d^{-s-i+k} \\ + (-1)^{n+k} \binom{-s+k}{n+1+k} \int_0^{+\infty} B_{n+k+1}(x)(x+d)^{-s-n-1} dx, $$

with $B_i(x)$ the i-th periodised Bernoulli polynomial Bernoulli, and $B_i$ the i-th Bernoulli number. We now have the following inequality

$$ |\zeta(s-k, d)| \le \frac{|d|^{1-\sigma+k}}{|1-s+k|} e^{|\tau|\pi/2} + \sum_{i=0}^{n+k} \left| \binom{-s+k}{i} \right| \frac{|B_{i+1}|}{i+1} |d|^{-\sigma-i+k} e^{|\tau|\pi/2} \\ + \left| \binom{-s+k}{n+k+1} \right| \int_0^{+\infty} |B_{n+k+1}(x)||x+d|^{-\sigma-n-1} e^{|\tau|\pi/2} dx. $$

Let's find an upper bound for each of these terms. We clearly have that $e^{|\tau|\pi/2} \le e^{r\pi/2}$. By compactness, we get that the continuous function $(x, y) \in [1/r', r'] \times [-r, r] \mapsto xy$ verifies the following inequality

$$ |d|^{-\sigma+k} \ll_{r,r'} r'^k. $$

By construction of the set $D(\delta, r)$, we have that

$$ \frac{|d|^{1-\sigma+k}}{|1-s+k|} \ll_{r,r'} \frac{r'^k}{\delta} \\ \ll_{r,r',\delta} r'^k. $$

We have the following inequality for the multinomial coefficients

$$ \forall i \in [0, n+k], \quad \left| \binom{-s+k}{i} \right| \le \binom{r+k+i-1}{i} \le \binom{2n+2k}{i}. $$

Let $i \in \mathbb{N}_0$. By [Leh40], we know that if $i \ne 2 \pmod 4$, we have

$$ \forall i \in \mathbb{N}_0, \forall x \in \mathbb{R}, \quad |B_i(x)| < \frac{2i!}{(2\pi)^i}, $$

and if we have $i = 2 \pmod 4$, we get

$$ \forall x \in \mathbb{R}, \quad |B_i(x)| < \frac{2i!\zeta(i)}{(2\pi)^i} \le \frac{2i!\zeta(2)}{(2\pi)^i}. $$

Let's note $\psi$ the digamma function. We know that $\lim_{x \to +\infty} \psi(x) = +\infty$. Therefore, there exist a real $x_0$ such that for all $x \ge x_0$, the fraction $\frac{\Gamma(x)}{(2\pi)^x}(\psi(x) - \ln(2\pi))$ is positive. We then find that the function

$$ x \in \mathbb{R}_+^* \mapsto \frac{\Gamma(x)}{(2\pi)^x} $$

is increasing after $x_0$. Therefore, for $n$ big enough, we have that

$$ \forall i \in [0, n+k], \quad \frac{i!}{(2\pi)^i} \ll \frac{(n+k)!}{(2\pi)^{n+k}} \ll \frac{1}{(2\pi)^k} k!(k+1)...(k+n). $$
Taking an upper bound of each term inside the product in the right hand side have the last equality, we get that $(k+1)...(k+n) \ll (k+1)^n$. Because $n = r + 1$, we then have

$$ \forall i \in [0, n+k], \forall x \in \mathbb{R}, \quad |B_i(x)| \ll \frac{k!}{(2\pi)^k} (k+1)^{r+1}. $$

By taking $x = 0$ in the previous inequality, we have

$$ \forall i \in [0, n+k], \quad |B_i| \ll \frac{k!}{(2\pi)^k} (k+1)^{r+1}. $$

By all the above, we have

$$ \begin{aligned} \sum_{i=0}^{n+k} \left| \binom{-s+k}{i} \right| \frac{|B_{i+1}|}{i+1} |d|^{-\sigma-i+k} & \ll |d|^{-\sigma+k} \frac{k!}{(2\pi)^k} (k+1)^n \sum_{i=0}^{2n+2k} \binom{2n+2k}{i} |d|^{-i} \\ & \ll \frac{k!}{(2\pi)^k} r'^k (k+1)^{r+1} (1+|d|^{-1})^{2n+2k} \\ & \ll \frac{k!}{(2\pi)^k} (k+1)^{r+1} (1+r')^{2k}. \end{aligned} $$

As $-\sigma - n \le -1$ it is clear that

$$ \forall x \in \mathbb{R}_+, \quad |x+d|^{-\sigma-n-1} \le (x+\mathrm{Re}(d))^{-\sigma-n-1}, $$

and the right hand side term is integrable over $\mathbb{R}_+$. Therefore, we obtain

$$ \left| \binom{-s+k}{n+k+1} \right| \int_0^{+\infty} |B_{n+k+1}(x)| |x+d|^{-\sigma-n-1} e^{i\tau|\pi/2}} dx \le \binom{n+2k+1}{n+k+1} \frac{2(n+k+1)!}{(2\pi)^{n+k+1}} \frac{\mathrm{Re}(d)^{-\sigma-n}}{\sigma+n}. $$

By a classical inequality, we have

$$ \binom{m}{i} \le e^i \left(\frac{m}{i}\right)^i \quad (0 \le i \le m). $$

We then find that

$$ \begin{aligned} \left| \binom{-s+k}{n+k+1} \right| \int_0^{+\infty} |B_{n+k+1}(x)| |x+d|^{-\sigma-n-1} e^{i\tau|\pi/2}} dx & \le 2e^k \left( \frac{n+2k+1}{k} \right)^k \frac{(n+k+1)!}{(2\pi)^{n+k+1}} \frac{\mathrm{Re}(d)^{-\sigma-n}}{\sigma+n} \\ & \le 2(2e)^k \left( 1 + \frac{n+1}{2k} \right)^k \frac{(n+k+1)!}{(2\pi)^{n+k+1}} \frac{\mathrm{Re}(d)^{-\sigma-n}}{\sigma+n}. \end{aligned} $$

From the following inequality

$$ \forall 0 \le x < k, \quad \left(1 + \frac{x}{k}\right)^k \le e^x, $$

we then find that

$$ \forall k > (n+1)/2, \quad \left(1 + \frac{n+1}{2k}\right)^k \le e^{(n+1)/2}. $$

As $n = r + 1$, we obtain

$$ \forall k \in \mathbb{N}_0, \quad \left(1 + \frac{n+1}{2k}\right)^k \ll 1. $$