数理解析研究所講究録
第 1806 巻 2012 年 123-133

123

Higher moments of the Epstein zeta functions
Keiju Sono

1

(University of Tokyo)

Introduction

Moments of the Riemann zeta function and other -functions have been studied
for about one hundred years, from the age of Hardy and Littlewood. Let
be the Riemann zeta function. In 1918, Hardy and Littlewood [2] investigated, andthe mean square (second moment) ofon the critical line
obtained the asymptotic formula$L$$\zeta(s)$$\zeta(\mathcal{S})$${\rm Re}(s)= \frac{1}{2}$(1.1)$\int_{1}^{T}|\zeta(\frac{1}{2}+it)|^{2}dt=T\log T+O(T)$as$Tarrow\infty$. Further, in 1926, Ingham [6] considered the fourth moment of$\zeta(\mathcal{S})$and proved that
(1.2)$l^{T}| \zeta(\frac{1}{2}+it)|^{4}dt=\frac{1}{2\pi^{2}}T(\log T)^{4}+O(T(\log T)^{3})$holds as $Tarrow\infty$ . The basic tools of them are the approximate functional
equations forand. Therefore, one might think that we can obtain
the asymptotic formula for the higher moments (sixth moment, eighth moment,by using the approximate functional
etc ) ofon the critical line
equations for $\zeta(s)^{k}(k\geq 3)$ . However, although these approximate functional
equations are known, a straightforward application of them doesn’t give thedesirable results. In fact, it is generally conjectured that$\zeta(s)^{2}$$\zeta(s)$$\cdots$$\zeta(s)$${\rm Re}(s)= \frac{1}{2}$(1.3)$\int_{1}^{T}|\zeta(\frac{1}{2}+it)|^{2k}dt\sim C_{k}T(\log T)^{k^{2}}$, but this has not been proved except
holds for all $k\geq 0$ with some constantfor the cases $k=0,1,2$ . Evaluating these moments is related to many topics inor theanalytic number theory, for example, the zero-density estimate fororder estimate foron the critical line. Also, the auther thinks this themeis sufficiently interesting in itself.
In this article 1, we consider the Epstein zeta function $\zeta(s;Q)$ , where is apositive definite symmetric matrix $(n\geq 4)$ which gives an integer-valued
lAlmost all parts of this article are some generalizations or summaries of the contents of$C_{k}$$\zeta(s)$$\zeta(s)$$Q$$n\cross n$

the auther’s another paper [11], dealing with the fourth moment of the Epstein zeta functions.

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quadratic form. We evaluate the moments of $\zeta(s;Q)$ on the line
and prove that the integral
is evaluated by $O(T(\log T)^{k^{2}})$
as $Tarrow\infty$ under the assumption of a moment conjecture for the Dirichlet Lis not the center of the functional
functions. Although the line$\zeta(s;Q)$, the auther thinks this formulation of problem is quite natequation ofural.
positive definite
Let us introduce the basic idea of this article. For asymmetric matrix , the quadratic form associated tois defined by $Q[x]=$$t_{xQx}$ for $x\in R^{n}$ . We assume that $Q[x]\in N$ for any
. For
$x\in
Z^{n}$
we defineby the number ofwhich satisfies $Q[x]=l$ . Then the
Epstein zeta function $\zeta(s;Q)$ is expressed by${\rm Re}(s)= \frac{n-1}{2},$$\int_{0}^{T}|\zeta(\frac{n-1}{2}+it;Q)|^{2k}dt$${\rm Re}(s)= \frac{n-1}{2}$$n\cross n$

$Q$

$Q$

$x\in Z^{n}\backslash \{0\}$$l\in Z_{\geq 0},$$r_{Q}(l)$(1.4)$\zeta(s;Q)=\sum_{\iota=1}^{\infty}\frac{r_{Q}(l)}{l^{s}}$for${\rm Re}(s)> \frac{n}{2}$. The corresponding theta series$\theta(z;Q)=\sum_{l=0}^{\infty}r_{Q}(l)e^{2\pi ilz}$

becomes a modular form of weight and decomposes into the sum of an Eisenstein series and a cusp form. Therefore, $\zeta(s;Q)$ decomposes into the sum of the-function associated to the Eisenstein series and the -function associated tothe cusp form. Hence to obtain the upper bound for the momens of $\zeta(s;Q)$ , itsuffices to evaluate the integrals of these two -functions. By using a classicalmethod in analytic number theory, we can prove that the moments of the Lfunction associated to the cusp form is evaluated by $O(T)$ , and our main problem
is to evaluate the moment of -function associated to the Eisenstein series. For
this purpose, we use the classical theories due to Hecke ([5]), Malyshev ([8]), andSiegel ([10]). By using their theorems, we prove that the -function associatedto the Eisenstein series is expressed by some finite or infinite series consistingof the Dirichlet -functions and thus we can obtain some upper bounds for the
moments of $\zeta(s;Q)$ by assuming a conjecture for the moments of the Dirichlet-functions.
As the easiest example, we take $Q=I_{4}$ , theunit matrix. Then the
Epstein zeta function
is expressed by$\frac{n}{2}$

$L$

$L$

$L$

$L$

$L$

$L$

$L$

$4\cross 4$$\zeta(s;I_{4})$$\zeta(s;I_{4})=8(1-2^{1-s})\zeta(s)\zeta(s-1)$

.

(1.5)

, the 2k-th
Since the factor $(1-2^{1-s})\zeta(s)$ is bounded on the line$o(T(\log T)^{k^{2}})$
momentis evaluated by
as $Tarrow\infty$ if we
assume that the conjecture (1.3) is valid. Of course, the general case is much
more complicated, but the underlying idea is similar. Among others, desirable
upper bounds for the fourth moment of $\zeta(s;Q)$ are obtained unconditionally,
since we have the unconditional results for the fourth moment of the Riemannzeta function or the Dirichlet -functions.
${\rm Re}(s)= \frac{3}{2}$$\int_{0}^{T}|\zeta(\frac{3}{2}+it;I_{4})|^{2k}dt$

$L$

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2Moments of Epstein zeta functions
2.1

Notation and some basic results

Let be a positive integer and be a
The Epstein zeta function associated to
$n$

$Q$

positive definite symmetric matrix.
is defined by$n\cross n$$Q$$\zeta(s;Q)=\sum_{x\in Z^{n}\backslash \{O\}}Q[x]^{-s} ({\rm Re}(s)>\frac{n}{2})$where $Q[x]$ $:=t_{xQx}$ . Like the Riemann zeta function, this function has themeromorphic continuation to the whole -plane and satisfies the following func$s$

tional equation:$\pi^{-s}\Gamma(s)\zeta(s;Q)=(\det Q)^{-\frac{1}{2}}\pi^{s-\frac{n}{2}}\Gamma(\frac{n}{2}-s)\zeta(\frac{n}{2}-s;Q^{-1})$

.

(2.1)

is holomorphic everywhere except for a simple pole atwith residue. Throughout this article, we assume that $Q[x]\in N$ for any. Let
be the number of $x\in Z^{n}$ which satisfies $Q[x]=l$ . Then$\zeta(s;Q)$ has the following
Dirichlet series expansion in:$\zeta(s;Q)$$s= \frac{n}{2}$$\pi^{\frac{n}{2}}/(\det Q)^{1}\Sigma\Gamma(\frac{n}{2})$$x\in Z^{n}\backslash \{0\}$$r_{Q}(l)$${\rm Re}(s)> \frac{n}{2}$$\zeta(s;Q)=\sum_{l=1}^{\infty}\frac{r_{Q}(l)}{l^{s}}.$

Hereafter, we assume that $n\geq 4$ . We consider the theta series corresponding
to $\zeta(s;Q)$ defined by$\theta(z;Q)=\sum_{l=0}^{\infty}r_{Q}(l)e^{2\pi ilz}$

It is known thata cusp form:$\theta(z;Q)$

is decomposed into the sum of an Eisenstein series and$\theta(z;Q)=E(z)+S(z)$

(2.2)

where
$E(z)= \sum_{l=0}^{\infty}e(l)e^{2\pi ilz}$

is the Eisenstein series and$S(z)= \sum_{i=1}^{\infty}s(l)e^{2\pi dz}$is the cusp form. Moreover, it is known that the coefficient $s(l)$ of $S(z)$ isevaluated by$s(l)\ll l^{\frac{n}{4}-\frac{1}{2}+\epsilon}$

if

$n$

(2.3)

is even, and$s(l)\ll l^{\frac{n}{4}-\frac{1}{4}+\epsilon}$

(2.4)

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is odd, where is always an arbitrary positive number throughout this article. Firstly, since the coefficient $s(l)$ is relatively small, the integral$it)|^{2k}dt$ also becomes relatively small. That is, by using a classical method inanalytic number theory, the following lemma is obtained:

if

$n$

$\epsilon$$\int_{0}^{T}|\hat{S}(\frac{n-1}{2}+$Lemma 2.1. When $Tarrow\infty$ , we have$\int_{0}^{T}|\hat{S}(\frac{n-1}{2}+it)|^{2k}dt=O(T)$

.

(2.5)

Thus our main problem is to evaluate the integral
For this purpose, we use the relations between the L–fUnction associated to the
Eisenstein series and the Dirichlet -functions. Let $L(s, \chi)$ be the Dirichlet Lfunction associated to a Dirichlet character . As an analogue of the moment
conjecture (1.3) for the Riemann zeta function, the following conjecture seems
to be natural:$\int_{0}^{T}|\hat{E}(\frac{n-1}{2}+it)|^{2k}dt.$$L$$\chi$

Conjecture 2.2. As$qarrow\infty,$$Tarrow\infty$, we have(2.6)$\sum_{\chi(modq)}\int_{1}^{T}|L(\frac{1}{2}+it, \chi)|^{2k}dt\ll qT(\log qT)^{k^{2}}$for any positive number . Here,characters modulo$k$$\sum_{\chi(}$modq) denotes thesum over all Dirichlet$q.$

Remark 2.3. In the book [9], Montgomery mentioned that the estimate(2.7)$\sum_{\chi(modq)}^{*}l^{T}|L(\frac{1}{2}+it, \chi)|^{4}dt\ll\emptyset(q)T(\logqT)^{4}$

denotes the sum over all primitive Dirichholds unconditionally. Here,let characters $mo$ dulo and denotes Euler’s -function. As an easy consequence of (2.7) (in detail, see [4]), the estimate (2.6) holds unconditionally inthe case of $k=2.$$\sum_{\chi(modq)}^{*}$$q$$\phi$$\phi$

The following lemma is famous hybrid bounds for Dirichlet
proved by Heath-Brown (see [3]):$L$-functions,Lemma 2.4. Let $L(s, \chi)$ be a Dirichlet -function associated to a Dirichlet, the following estimates hold:character modulo . Then. when
$L$

$q$

$tarrow\infty$

l,

$t$ @
$\log(qt)1$$L( \frac{1}{2}+it, \chi)\ll q^{2}$$L( \frac{1}{2}+it, \chi)\ll(qt)^{-}1B^{+\epsilon}3$

Further, we prepare the following inequality:

.

(2.8)
(2.9)

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Lemma 2.5. For$k \geq\frac{1}{2}$and$x_{1},$$\cdots,$$x_{m}\geq 0_{f}$we have$x^{\frac{1}{12k}}+\cdots +x^{\frac{1}{m2k}}\leq m^{1-\frac{1}{2k}}(x_{1}+\cdots+x_{m})^{\frac{1}{2k}}$.

(2.10)

Proof. The inequality (2.10) is equivalent to$\frac{x^{\frac{1}{1^{2k}}}+\cdots+x^{\frac{1}{m2k}}}{m}\leq(\frac{x_{1}+\cdots+x_{m}}{m})^{2}\pi^{1}$and we can easily prove this inequality by using the convexity of the function$f(x)=x^{\frac{1}{2k}}.$$\square$Now the main theorem is stated as follows:
Theorem 2.6. Assume that is even and $n\geq 4$ , orunder the assumption of the Conjecture 2.2, for
$n$

$n$

$k \geq\frac{1}{2}$is odd and $n\geq 7$ . Then,, the following estimate

holds;$\int_{0}^{T}|\zeta(\frac{n-1}{2}+it;Q)|^{2k}dt=O(T(\log T)^{k^{2}})$

Proof. Firstly, we assume that.

(2.11)

is even and $n\geq 4$ . Then, the Eisenstein series
is a modular form of weight and level $N$ , where $N$ is a positive integer
such that $NA^{-1}$ becomes the integral matrix for $A=2Q$ (see [7]). According
to Hecke’s paper [5], the series
is expressed by some linear combination of
the form$n$$\hat{E}(z)$$\frac{n}{2}$$\hat{E}(s)$$(t_{1}t_{2})^{-s}L(s, \chi_{1})L(s-\frac{n}{2}+1, \chi_{2})$whereare positive divisors of level $N$ andare Dirichlet charac, respectively. Sinceters modulo
is bounded on the, and since the Conjecture 2.2 indicates that each integral
line
is evaluated by $o(T(\log T)^{k^{2}})$ , by applying Minkowski’s
inequaliy, the 2k-th moment ofis also evaluatedon the line
by $O(T(\log T)^{k^{2}})$ . Therefore, the statement of theorem is proved in this case.
Next, we assume thatis odd and $n\geq 7$ . The computations below is a
simple arrangement of the Fomenko’s technique introduced in [1]. In this case,
Malyshev, about fifty years ago, showed that the Fourier coefficient $e(l)$ of theEisenstein series $E(s)$ has the following expression (see [8]):$t_{1},$ $t_{2}$$\chi_{1},$$\frac{N}{t_{1}},$$\chi_{2}$$(t_{1}t_{2})^{-s}L(s, \chi_{1})$$\frac{N}{t_{2}}$${\rm Re}( \mathcal{S})=\frac{n-1}{2}$$\int_{0}^{T}|L(\frac{1}{2}+it, \chi_{2})|^{2k}dt$$\hat{E}(s)$${\rm Re}(s)= \frac{n-1}{2}$$n$$e(l)= \frac{\pi^{\frac{n}{2}}}{(\det Q)^{\frac{1}{2}}\Gamma(\frac{n}{2})}l^{\frac{n}{2}-1}H(Q;l)$where$H(Q;l)= \sum_{q=1}^{\infty}\{\sum_{h(mod \acute{q})}q^{-n}S(hQ;q)e^{-2\pi i\frac{lh}{q}}\}$

128
is a singular series,$\sum’$

means the sum over a reduced residue system, and$S(Q;q)= \sum_{\rangle}^{q.-1}ex_{1},\cdot\cdot x_{n}=0\frac{2\pi iQ(x_{1},\cdots,x_{n})}{q}$is a Gaussian sum. Therefore, the associated Dirichlet series is given by$\hat{E}(s)=\frac{\pi^{n}\tau}{(\det Q)^{\frac{1}{2}}\Gamma(\frac{n}{2})}\sum_{l=1}^{\infty}\frac{1}{l^{s-\tau+1}n}\sum_{q=1}^{\infty}\sum_{h(mod \acute{q})}q^{-n}S(hQ;q)e^{-2\pi i\frac{lh}{q}}$for ${\rm Re}(s)>$$(q_{1}, m)=1$;. Let $(l, q)=d,$$l=k_{1}d,$ $q=q_{1}d,$ $(k_{1}, q_{1})=1$. Then the right hand side becomes

and $k_{1}=k_{2}q_{1}+m,$$\frac{\pi^{\frac{n}{2}}}{(\det Q)^{\frac{1}{2}}\Gamma(\frac{n}{2})}\sum_{d=1}^{\infty}\frac{1}{d^{s-?^{+1}}n}\sum_{q_{1}=1}^{\infty}\sum_{h(modq_{\acute{1}}d)}(q_{1}d)^{-n}S(hQ;q_{1}d)$$\sum_{m(mod \acute{q}_{1})}e^{-\frac{2\pi ibm}{q_{1}d}}\sum_{k_{1}\equiv m(modq_{1})}\frac{1}{k_{1^{-z+1}}^{s^{n}}}.$The last sum above is rewritten by using Dirichlet -functions. By applying thewell-known identity$L$$\sum_{\chi(modq_{1})}\overline{\chi}(m)\chi(l)=\{\begin{array}{l}\phi(q_{1}) (l\equiv m (modq_{1}))0 (otherwise),\end{array}$we have$\sum_{k_{1}\equiv m(modq_{1})}\frac{1}{k_{1^{-\tau+1}}^{s^{n}}}=\frac{1}{\phi(q_{1})}\sum_{\chi(modq_{1})}\overline{\chi}(m)\sum_{k_{1}=1}^{\infty}\frac{\chi(k_{1})}{k_{1}^{s-@+1}}$$= \frac{1}{\phi(q_{1})}\sum_{\chi(modq_{1})}\overline{\chi}(m)L(s-\frac{n}{2}+1, \chi)$for${\rm Re}(s)> \frac{n}{2}$. Therefore,$\hat{E}(s)=\frac{\pi^{\frac{n}{2}}}{(\det Q)^{\frac{1}{2}}\Gamma(\frac{n}{2})}\sum_{d=1}^{\infty}\frac{1}{d^{s-\frac{n}{2}+1}}\sum_{q_{1}=1}^{\infty}\sum_{h(modq_{\acute{1}}d)}\frac{S(hQ;q_{1}d)}{(q_{1}d)^{n}}$(2.12)$\sum_{m(mod \acute{q}_{1})}e^{-\frac{2\pi ihm}{q_{1}}}\frac{1}{\phi(q_{1})}\sum_{\chi(modq_{1})}\overline{\chi}(m)L(s-\frac{n}{2}+1, \chi)$holds for${\rm Re}(s)> \frac{n}{2}$. It is known that the following estimate holds (see [8]):$S(hQ;q)\ll q^{\frac{n}{2}}.$

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The estimate above is not dependent on . Therefore, the absolute value of theright hand side of (2.12) is estimated by$h$$\ll\sum_{d=1}^{\infty}\frac{1}{d^{\sigma-\frac{n}{2}+1}}\sum_{q_{1}=1}^{\infty}\phi(q_{1}d)\frac{(q_{1}d)^{\frac{n}{2}}}{(q_{1}d)^{n}}\cdot\phi(q_{1})\frac{1}{\phi(q_{1})}\sum_{\chi(modq_{1})}|L(s-\frac{n}{2}+1, \chi)|$$\ll\sum\frac{1}{d^{\sigma}}\infty\sum^{\infty}\frac{1}{\frac{n}{2}-1}$$\sum$$|L(s- \frac{n}{2}+1, \chi)|.$$d=1 q_{1}=1q_{1} \chi(modq_{1})$(2.13)The estimate (2.9) yields the right hand side of (2.13) converges on the line
, henceis continued analytically to some domain containing
the lineby (2.12) and the estimate$\hat{E}(s)$${\rm Re}(s)= \frac{n-1}{2}$${\rm Re}(s)= \frac{n-1}{2}$$| \hat{E}(\frac{n-1}{2}+it)|\ll\sum_{q_{1}=1}^{\infty}\frac{1}{q_{1}^{Z^{-1}}n}\sum_{\chi(modq_{1})}|L(\frac{1}{2}+it,\chi)|$(2.14)

holds. By applying Minkowski’s inequality to (2.14), we have$( \int_{0}^{T}|\hat{E}(\frac{n-1}{2}+it)|^{2k}dt)^{\pi^{1}}$(2.15)$\ll\sum_{q_{1}=1}^{\infty}\frac{1}{q^{\frac{n}{12}-1}}\sum_{\chi(mod q_{1})}(\int_{0}^{T}|L(\frac{1}{2}+it, \chi)|^{2k}dt)^{\pi^{1}}$By applying the inequality (2.10) to the sum in $\chi(modq_{1})$ and using the estimate(2.6), the right hand side of (2.15) is evaluated by$\leq\sum_{q_{1}=1}^{\infty}\frac{1}{q^{\frac{n}{12}-1}}\phi(q_{1})^{1-\frac{1}{2k}}(\sum_{modq_{1}}\int_{0}^{T}|L(\frac{1}{2}+it, \chi)|^{2k}dt)^{\pi^{1}}$$\ll\sum_{q_{1}=1}^{\infty}\frac{1}{q^{\frac{n}{12}-1}}q_{1}^{1-\frac{1}{2k}}(q_{1}T(\log q_{1}T)^{k^{2}})^{\frac{1}{2k}}$$\ll(\sum_{q_{1}=1}^{\infty}\frac{1}{q^{\frac{n}{12}-2-\epsilon}})T^{\frac{1}{2k}}(\log T)^{\frac{k}{2}}.$The series$\sum_{q_{1}=1}^{\infty}\frac{1}{q_{l}g-2-\epsilon}$

converge when $n>6$ . Therefore, the estimate$( \int_{0}^{T}|\hat{E}(\frac{n-1}{2}+it)|^{2k}dt)^{\frac{1}{2k}}\ll T^{\pi^{1}}(\log T)^{k}z$holds when $n\geq 7$ , hence the statement of theorem is proved.

$\square$130
Next, we consider the case of $n=5$ . In this case, we cannot use the method
we used in the proof of Theorem 2.6, since the right hand side of $(2.i3)$ may
not converge on the line ${\rm Re}(s)=2$ in the case of $n=5$ . To obtain the upper, we use another formula proved by Siegel ([10])bound for the moments ofunder some additional conditions.
positive definite symmetric integer matrix which
Theorem 2.7. Let be a
, under the assumption of the Conjecture
$\det
Q=1$
. Then, forsatisfies
2.2, we have$\hat{E}(s)$$5\cross 5$$Q$$k> \frac{1}{2}$(2.16)$\int_{0}^{T}|\zeta(2+it;Q)|^{2k}dt=O(T(\log T)^{k^{2}})$as$Tarrow\infty.$

satisfies the conditions of theorem. In this case, Siegel
has the following expression (see [10], Theorem 12):

Proof. Assume thatshowed that$\hat{E}(s)$$Q$(2.17)$\hat{E}(s)=2\pi^{s}\frac{\Gamma(\frac{5}{2}-s)}{\Gamma(\frac{5}{2})}\{\psi(s)+\psi(\frac{5}{2}-s)\}$for$1<{\rm Re}( \mathcal{S})<\frac{3}{2}$, where the function$\psi(s)$is defined by$\psi(s)=2^{s-5}\mathfrak{T}\{$$\cos\frac{\pi}{4}(2s-5)_{a,b}\sum_{b\equiv 1(mod4)}\chi_{b}(a)a^{s-\frac{6}{2}}b^{-s}$(2.18)$+\cos$ $\frac{\pi}{4}(2s+5)_{a,b}\sum_{b\equiv 3(mod4)}\chi_{b}(a)a^{s-\S}b^{-s}\}$and$\chi_{b}(a)=(\frac{a}{b})$denoting the Legendre-Jacobi symbol. For fixed , we have$b$$\sum_{a}\chi_{b}(a)a^{e-\frac{6}{2}}=L(\frac{5}{2}-s, \chi_{b})$for${\rm Re}(s)< \frac{3}{2}$. Therefore,$a,b \sum_{b\equiv j(mod4)}\chi_{b}(a)a^{s-\frac{5}{2}}b^{-s}=\sum_{b\equiv j(mod4)}b^{-s}L(\frac{5}{2}-s, \chi_{b})$(2.19). By using the estimate (2.8), the series of theholds forright hand side of (2.19) converge absolutely on ${\rm Re}(s)=2$ , so the left handside of (2.19) can be continued analytically to some domain containing the line${\rm Re}(s)=2$ by (2.19). Therefore,
can be continued analytically to somedomain containing the line ${\rm Re}(s)=2$ by$(j=1,3)$${\rm Re}(s)< \frac{3}{2}$$\psi(s)$$\psi(s)=2^{s-i}2\{$ $\cos\frac{\pi}{4}(2s-5)\sum_{b\equiv 1(mod4)}b^{-s}L(\frac{5}{2}-s, \chi_{b})$(2.20)$+\cos$ $\frac{\pi}{4}(2s+5)\sum_{b\equiv 3(mod4)}b^{-s}L(\frac{5}{2}-s, \chi_{b})\}.$

131
On the other hand, for fixed$a,$$\sum_{b,b\equiv j(mod4)}\chi_{b}(a)b^{-s}$$= \frac{1}{\phi(4)}\sum_{\chi(mod4)}\overline{\chi}(j)\sum_{b=1}^{\infty}\chi(b)\chi_{b}(a)b^{-s}$$= \frac{1}{\phi(4)}\sum_{\chi(mod4)}\overline{\chi}(j)L(s,\tilde{\chi}_{a,\chi})$$(j=1,3)$ holds for ${\rm Re}(s)>1$ , where$\tilde{\chi}_{a,\chi}(b)=\chi(b)\chi_{b}(a)=\chi(b)(\frac{a}{b})$.

(2.21)

By a straightforward exercise, we can prove that
becomes a Dirichlet character modulo $4a$ . Therefore, we have proved that the identity$\tilde{\chi}_{a,\chi}$$\psi(s)=\frac{2^{s-\frac{5}{2}}}{\phi(4)}\{$$\cos\frac{\pi}{4}(2s-5)\sum_{a=1}^{\infty}a^{s-\frac{5}{2}}\sum_{\chi(mod4)}\overline{\chi}(1)L(s,\tilde{\chi}_{a,\chi})$(2.22)$+$ $\cos\frac{\pi}{4}(2s+5)\sum_{\alpha=1}^{\infty}a^{s-\frac{5}{2}}\sum_{\chi(mod4)}\overline{\chi}(3)L(\mathcal{S},\tilde{\chi}_{a,\chi})\}$

holds for $1<{\rm Re}(s)< \frac{3}{2}$ , whereis a Dirichlet character modulo $4a$ . By
using Heath-Brown’s estimate (2.8) again, the right hand side of (2.22) converges, soabsolutely at
can be continued analytically to some domain
containing the lineby the identity (2.22). Therefore, by combining
these results, the -function
has the following Dirichlet series expansion${\rmRe}(s)=2$
on the line:$\tilde{\chi}_{a,\chi}$$s= \frac{1}{2}+it$$\psi(s)$${\rm Re}(s)= \frac{1}{2}$$L$$\hat{E}(s)$$\hat{E}(2+it)$$=2^{\frac{1}{2}-it} \pi^{2+it^{\Gamma(\frac{1}{\Gamma 2}-it)}}(\frac{5}{2})\{\cos\frac{\pi}{4}(-1+2it)\sum_{b\equiv 1(mod4)}b^{-2-it}L(\frac{1}{2} -- it, \chi_{b})$$+ \cos\frac{\pi}{4}(9+2it)\sum_{b\equiv 3(mod4)}b^{-2-it}L(\frac{1}{2} -- it, \chi_{b})\}$$+2^{-2-it} \pi^{2+it^{\Gamma(\frac{1}{r^{2}}-it)}}(\frac{5}{2})\{\cos\frac{\pi}{4}(-4-2it)\sum_{a=1}^{\infty}a^{-2-it}\sum_{\chi(mod4)}\overline{\chi}(1)L(\frac{1}{2}-it,\tilde{\chi}_{a,\chi})$$+ \cos\frac{\pi}{4}(6-2it)\sum_{a=1}^{\infty}a^{-2-it}\sum_{\chi(mod4)}\overline{\chi}(3)L(\frac{1}{2} -- it, \tilde{\chi}_{a,\chi})\}.$(2.23)

132
Note that
Stirling’s formula). Now, for$\Gamma(\frac{1}{2}-it)\cos\frac{\pi}{4}(\cdot\pm 2it)$( terms) are bounded when $tarrow\infty$ (use, by applying Minkowski’s inequality, we$4$$k> \frac{1}{2}$have$( \int_{0}^{T}|\hat{E}(2+it)|^{2k}dt)^{\pi^{1}}$$\ll\sum_{b\equiv 1(mod4)}b^{-2}(\int_{0}^{T}|L(\frac{1}{2} -- it, \chi_{b})|^{2k}dt)^{\frac{1}{2k}}$$+ \sum_{b\equiv 3(m\circ d4)}b^{-2} (\int_{0}^{T}|L (\frac{1}{2} -- it, \chi_{b})|^{2k}dt)^{\overline{2}7}1$$+ \sum_{a=1}^{\infty}a^{-2}(\int_{0}^{T}|L(\frac{1}{2} -- it, \tilde{\chi}_{a,\chi})|^{2k_{dt)^{2}}\pi^{1}}$$\ll\sum_{b\equiv 1,3(mod4)}b^{-2}(bT(\log bT)^{k^{2}})$rk$+ \sum_{a=1}^{\infty}a^{-2}(aT(\log aT)^{k^{21}})\overline{2}F$$\ll T^{\pi^{1}}(\log T)^{k}\mathfrak{B}.$

Therefore, the estimate$\int_{0}^{T}|\hat{E}(2+it)|^{2k}dt\ll T(\log T)^{k^{2}}$holds. Thus we obtain the estimate (2.16).

$\square$Since the estimate (2.6) in Conjecture 2.2 holds unconditionally in the caseof $k=2$ , as a corollary of Theorem 2.6 and Theorem 2.7, we obtain the following
result for the fourth moment of $\zeta(s;Q)$ :Corollary 2.8. Unconditionally, for any
Theorem 2.6 or Theorem 2.7, we have$n\cross n$

positive definite matrix$\int_{0}^{T}|\zeta(\frac{n-1}{2}+it;Q)|^{4}dt\ll T(\log T)^{4}$

as

3

$Q$

in

(2.24)$Tarrow\infty.$

Acknowledgement

The auther would like to express his gtatitude to Professor Takumi Noda, who
was the organizer of the RIMS symposium in 2011, for giving the opportunity
to talk about this topic. He also thanks many people who gave him a lot of
valuable advices at the symposium.

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Graduate school of Mathematical Sciences,
University of Tokyo,
Komaba, Meguro,
Tokyo, Japan