数理解析研究所講究録
1091 巻 1999 年 251-255

251

Riemann ゼータ関数の近似関数等式に対する平均値公式

1

山口大学理学部

木内功

中央情報開発 (株)柳沢直樹(Isao Kiuchi)(Naoki Ymagisawa)

Statement of result.
Let$R_{1}(s)$be the Riemann zeta-function, and define the remainder termin the approximate functional equation forby$\zeta(s)$$\zeta(s)$$R_{1}(s \rangle=\zeta(s)-n\sum_{\leq\sqrt{\frac{t}{2\pi}}}\frac{1}{n^{s}}-\chi(S)n\leq\sqrt{\mathrm{R}^{t}}\sum\frac{1}{n^{1-s}}$where$\chi(s)=2^{s}\pi^{S}-1\sin(\frac{1}{2}\pi s)\Gamma(1-\mathit{8})$.

(1)

The aim of this note is to derive the $2k$-th power moments of the function$|R_{1}(s)|$ in the critical strip $0\leq\sigma\leq 1$ .
We can prove the following

Theorem [1]. For any positive integers , we have$k$$\int_{1}^{T}|R1(S)|^{2}kdt$$\frac{(2\pi)^{k\sigma}C_{k}}{1-k\sigma}T^{1-k\sigma}+\mathrm{Y}_{k,\sigma}(T)$$=\{$$\frac{(2\pi)^{k\sigma}C_{k}}{1-k\sigma}T^{1-k\sigma}+D_{k,\sigma}+\mathrm{Y}_{k,\sigma}(T)$if$0 \leq\sigma\leq\frac{1}{2k}$if$\frac{1}{2k}<\sigma\leq 1$

if$2\pi C_{k}\log T+D_{k_{\mathrm{E}}},1+\mathrm{Y}_{k,\not\in}(T)$

(2)

and

$\sigma\neq\frac{1}{k}$

(3)
(4)

$\sigma=\frac{1}{k}$

urith
(5)

$\mathrm{Y}_{k,\sigma}(T)=O(T3-k\sigma)$

where the constant$D_{k,\sigma}$

depends on$k$and , and$\sigma$$C_{k}= \int_{0}^{1}(\frac{\cos(2\pi(y-y-2\frac{1}{16}))}{\cos(2\pi y)})^{\mathit{2}k}d?d$

.

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Remark. This theorem indude8 the fact that$\Omega(t^{-\frac{\sigma}{2}})$$R_{1}(s)=\{$$\Omega(t^{-\frac{1}{2}}(\log t)^{\frac{1}{2}})$

2

if
if

$0\leq\sigma<1$$\sigma=1$,.

Proof of the formula(2).

To prove our theorem, we start with the weak form of the “Riemann-Siegel, we have
: Fbrformula” for$\zeta(s)$$0\leq\sigma\leq\perp$$x(1-S)^{\frac{1}{2}R_{1(}}s)=(-1)[ \sqrt{\frac{t}{2\pi}}]-1\frac{\cos(2\pi(\delta 2-\delta-\frac{1}{16}))}{\cos(2\pi\delta)}(\frac{t}{2\pi})^{-\frac{1}{4}}+O(t^{-3}4)$

where$\delta=\sqrt{\frac{t}{2\pi}}-[\sqrt{\frac{t}{2\pi}]}$assume thatwith $[x]$ being the integer part of. Rom the above, we have, for$T_{1}<T_{2}\leq 2T_{1}$$x$(see [2]). We$0 \leq\sigma\leq\frac{1}{2k}$,$\int_{T_{1}}^{T_{2}}|R_{1}(s)|^{2}dt=I1(k\tau 1,T_{2})+O(_{j1}\sum_{=}^{2k}|I1(\tau_{1},\tau 2)|^{1-}*|I2(T_{1},T2)|^{*})$

,

(6)

where
$I_{1}(T_{1},T2)= \int_{T_{1}}|x(s)|k(\tau_{2}\frac{t}{2\pi})^{-}\frac{k}{2}(\frac{\cos(2\pi(\delta 2-\delta-\frac{1}{16}))}{\cos(2\pi\delta)})^{2k}dt$

and$I_{2}(T_{1}, \tau_{2})=\int_{T_{1}}^{\tau_{2}}t^{-\frac{3k}{2}}|\chi(S)|dtk$.

By using the asymptotic formula of (1), we have(7)$(t>0)$$| \chi(S)|^{k}=(\frac{t}{2\pi})^{k(_{2^{-\sigma}}})\sigma\iota+ck,(t)$where$G_{k,\sigma}=O(t^{k()-1} \frac{1}{2}-\sigma)$.

(8)

It is easily seen that
$I_{2}(T_{1},T2)=O(T_{1}^{1-}k-k\sigma)$.

(9)

Rom (7) and (8), we get$I_{1}(T_{1,2}\tau)=I1,1(T_{1},\tau_{2})+o(T_{1}^{-1}|I1,1(T_{1},\tau 2)|)$

,(10)

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where
$I_{1,1}(T_{1,\mathrm{z}} \tau)=\int_{\tau}1\tau_{2}(\frac{t}{2\pi})^{-k\sigma}(\frac{\cos(2\pi(\delta 2-\delta-\frac{1}{16}))}{\cos(2\pi\delta)})^{2k}dt$Let$N_{1}$

be the smallest integer such that

integer such that

$N_{2}\leq\sqrt{\frac{T}{2}\pi 2}$$\sqrt{\frac{T}{2}\pi 1}\leq N_{1}$. We obtain, forand$0 \leq\sigma\leq\frac{1}{\mathit{2}k}$$N_{2}$.

the largest,$I_{1,1}(T_{1},\tau_{2})$$=4 \pi\int_{0}^{1}(\frac{\cos(2\pi(y-2y-\frac{1}{16}))}{\cos(2\pi y)})^{2k}N211\sum_{n=N1}^{-}(y+n)^{1}-2k\sigma dy+o(T^{1}\mathrm{Z}-k\sigma)$$= \frac{2\pi}{1-k\sigma}\int_{0}^{1}(\frac{\cos(2\pi(y-2y-\frac{1}{16}))}{\cos(2\pi y)})^{2k}\{(y+\sqrt{\frac{T_{2}}{2\pi}})^{\mathit{2}2k\sigma}--(y+\sqrt{\frac{T_{1}}{2\pi}})^{2-\mathit{2}k}\sigma \mathrm{I}^{dy}$$+O(T_{1}^{\frac{1}{2}-k\sigma})$$= \frac{(2\pi)^{k\sigma}C_{k}}{1-k\sigma}(\tau_{2}^{1}-k\sigma-\tau 1-k\sigma)1+o(T_{1}^{\frac{1}{2}-})k\sigma$.

(11)

Substituting (9), (10) and (11) into (6), we obtain the formula (2).

Proof of the formulas (3) and (4).

3’From Lemma 1 in [1], we have, for any positive integers
,

$k$

and

$\frac{1}{2k}<\sigma\leq 1$. $\int_{1}^{T}|R_{1}(s)|^{2k}dt=J_{1}(1,T)+J_{2}(1,T)$ ,(12)where$J_{1}(T_{1},T_{\mathit{2}})= \int_{T_{1}}^{T_{2}}|\chi(s)|^{k}|\chi(\frac{1}{2k}+it)|^{-k}|R_{1}(\frac{1}{2k}+it)|^{2k}dt$

and$J_{2}( \tau_{1},T2)=\sum_{j=1}k\int_{\tau_{1}}^{\tau}2||\chi(\mathit{8})k|\chi(\frac{1}{2k}+it)|^{-}\{k-\mathrm{j})|R_{1}(\frac{1}{2k}+it)|2(k-\mathrm{j})jFk(t)dt$

with$F_{k}(t)=O(t^{-\frac{3}{4}}| \chi(\frac{1}{2k}+it)|^{-_{\mathrm{Z}}^{1}}|R_{1}(\frac{1}{2k}+it)|+t^{-\S)}$

.

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Applying (2),(7) $,(8)$ and integration by parts, we have, for $k\neq 2$ ,$\int_{T_{1}}^{T_{2}}|\chi(\frac{1}{2k}+it)|^{-k}|R_{1}(\frac{1}{2k}+it)|^{2k}dt$$= \frac{4\pi}{2-k}C_{k}(\frac{t}{2\pi})1-\frac{k}{2}+(\frac{t}{2\pi})^{\frac{1}{2}-\frac{k}{2}}\mathrm{Y}_{k,*}(t)|\tau_{1}\tau_{2}$$+ \int_{T_{1}}^{T_{2}}H_{k()}t|R_{1}(\frac{1}{2k}+it)|^{2k}dt+\frac{k-1}{2}(2\pi)^{\frac{k-1}{2}}\int_{T_{1}}^{T}2t^{-}\frac{1+k}{2}\mathrm{Y},1k_{\overline{2}T}(t)dt$

and for $k=2$ ,$\int_{T_{1}}^{T_{2}}|\chi(\frac{1}{4}+it)|^{-2}|R_{1}(\frac{1}{4}+it)|^{4}dt$$=2 \pi C_{2}\log t+\sqrt{2\pi}t^{-9_{\mathrm{Y}_{\mathit{2},\frac{1}{4}}}}(t)|_{T_{1}}^{T_{2}}+\int_{T_{1}}^{T_{2}}H_{2()}t|R_{1}(\frac{1}{4}+it)|^{4}dt$$+ \sqrt{\frac{\pi}{2}}\int_{T_{1}}^{T}2tt-_{2}3\mathrm{Y})2,\frac{1}{4}(dt$.

From (5),(7) and (8), we have$\int_{1}^{T}|\chi(\frac{1}{2k}+it)|^{-k}|R_{1}(\frac{1}{2k}+it)|^{2k}dt$$\frac{4\pi}{2-k}C_{k}(\frac{T}{2\pi})^{1-_{\mathfrak{T}}^{k}}$if $k\geq 1$ and $k\neq 2$$2\pi C_{2}\log T$

if $k=2$$=A_{k}+B_{k}(\tau)+\{$

with$B_{k}( \tau)=O(\tau\frac{1-k}{2})$where the constantandfor$\frac{1}{2k}<\sigma\leq 1$$A_{k}$,

depends on . Hence, integrating by parts, we obtain,,$k$$\sigma\neq\frac{1}{k}$$J_{1}(\tau_{1},\tau_{2})=o(T_{1}^{1-k\sigma})$and$J_{1}(1,T)= \frac{(2\pi)^{k\sigma}C_{k}}{1-k\sigma}T^{1-k\sigma}+L_{k,\sigma}+O(T^{1}2^{-}k\sigma)$with a certain constant$L_{k,\sigma}$. Similarly in case$\sigma=\frac{1}{k}$(13), we have$J_{1}(T_{1,2}\tau)=O(1)$and$J_{1}(1,T)=2\pi C_{k}\log T+L_{k_{\mathrm{E}}^{1}},+O(\tau^{-\frac{1}{2}})$

.

(14)

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By using H\"older’s inequality and the above, we have, for$J_{2}(1,T)=j_{2()\mathit{0}}1, \infty+(\tau\frac{1}{2}-k\sigma)$.

$\frac{1}{2k}<\sigma\leq 1$,(15)

Substituting (13), (14) and (15) into (12), we obtain the formulas (3) and(4).

References[1] I. Kiuchi and N. Yanagisawa, On the mean value formulas for the approximate functional equation of the Riemann zeta-function, preprint.
[2] C.L.Siegel, \"Uber Riemanns Nachlass zur analytischen Zahlentheorie,Quellen und Studien zur Geschichte der Math. Astr. und Physik, Abt.
B, Studien 2(1932), 45-80.