石川, 秀明

Kyoto University Research Information Repository

Title$L(1/2+it,\chi_j)$と$L(1/2+it,\chi_k)$の値の差について　(解析的整数論とその周辺)Author(s)石川, 秀明Citation数理解析研究所講究録 (2004), 1384: 239-248Issue Date2004-07URL http://hdl.handle.net/2433/25758RightType Departmental Bulletin PaperTextversionpublisherKyoto University238$L$ ( $1/2+it,$ $\chi$j) $L$ ( $1/2+ib,$ $\chi$k) 1(Hideaki Ishikawa)Graduate School of Science and Technology Niigata University1 Introduction$L$$L(s, \chi)=\sum_{n=1}^{\infty}\frac{\chi(n)}{n^{s}}$$s$ $\chi$ $\chi$ $\chi 0$$\Re s>0$$\zeta(s)$ zero$\Re s=1/2$$\Re s=1/2$$|L$ ( $1/2+it,\chi$j)| $|L(1/2+it,\chi k)|$ $1^{\mathrm{a}}$$|L$ ( $1/2+it,\chi$j)| $|L$ ( $1/2+it,$ $\chi$k)|:$\Lambda(T,\chi_{\mathrm{j}},\chi_{k})=\int_{0}^{T}|$L(1/2 $+i$t, $\chi j$ ) $|^{2}-|L(1/2+it, \chi_{k})|^{2}dt$( $\Lambda(T)$ ) $\text{ }Tarrow\infty$? $\Lambda(T)$:Theorem 1 Assume that $\chi j$ and $\chi k$ are primitive chamcters modulo $q$ and $\chi j\neq\chi k$ . Thenthere excist &(q), $c_{1}(q),$ $c_{2}(q)$ such $that_{f}$ for each $T>T0$ theoe exist $t_{1},$ $t_{2}\in[T, T+c_{2}\sqrt{T}]$satisfying $\Lambda(t1)>c_{1}t_{1}^{1/4}$ and $\Lambda(t_{2})<-$ c1t21/4.Remark 1 Unfortunately we can not make explicit the dependency of To, $c_{1}$ and $c_{2}$ withrespect to $q$ .Cororally 1 Let $\chi j$ and $\chi_{k}$ be as in Theorern 1. Then we have$\mathrm{A}(T)=\Omega\pm(T^{1/4})$ .In particular$\int_{0}^{T}|L(1/2+it,\chi 1^{2}dt$ $- \int_{-T}^{0}|$L $(1/2+it, \chi)|^{2}dt=\Omega\pm(T^{1/4})$holds.1384 2004 239-2482402 Theorem 1$L(s, \chi)$$I(T, \chi):=\int_{0}^{T}|$L $(1/2+it, \chi)|^{2}$dt.:$I(T,\chi)\sim M(T,q)$ $Tarrow\infty$ (1)$M$ (T, $q$) $= \frac{\phi(q)}{q}\{$T $\log T+T(\log\frac{q}{2\pi}+2\gamma-1+2\sum_{\mathrm{p}1q}\frac{1\mathrm{o}\mathrm{g}p}{p-1})\}$$\phi(q)$ Euler $\gamma$ Euler $p$ $q$$I$ (T, $\chi$) $-M$(T, $q$) $E$(T, $\chi$) $E$ (T, $\chi$)(1) $\mathrm{m}\mathrm{o}\mathrm{d} q$ $\chi j\text{ }\chi$k $I(T, \chi j)_{\text{ }}I$ (T, $\chi k$ )$\Lambda(T)$ $E$ (.T, $\chi j$ ) $-E$(T, $\chi_{k}$ ) $\Lambda(T)$( $(s)$ $E$ (T) [2], [8],[4] $E$ (T, $\chi$) T\rightarrow$E$ (T) $E$ (T, $\chi j$ ) $-E$(T, $\chi k$ )$?E(T, \chi j)$ $E$ (T, $\chi_{k}$ )?( ) $E(T,\chi)$ $E(T)$$E$ (L $\chi$) (Proposition $\mathfrak{o}$ $\Lambda(T)$T Theorem 13 Theorem 1$E$ (T, $\chi$):$e(T, u)$ $=$ $(1+ \frac{\pi u}{2T})^{-1/}4$ $($5ar $\mathrm{s}$in$\mathrm{h}\sqrt{\frac{\pi u}{2T}})-1$ ,$f(T, u)$ $=$ $2T\mathrm{a}\mathrm{r}\sinh\sqrt{\frac{\pi u}{2T}}+\sqrt{2\pi uT+\pi^{2}u^{2}}-\pi/4,$$g(T, u)$ $=$ $T \log\frac{T}{2\pi u}-T+2\pi u+\frac{7\Gamma}{4}$ ,$l(T,u)$ $=$ $\frac{T}{2\pi}+\mathit{2}$ $-\sqrt{\frac{u^{2}}{4}+\frac{uT}{2\pi}}$ ,$a(n,\chi)$ $=$ $\frac{1}{q}\sum_{k|n}\sum_{a=1}^{q}\chi$ (a)$\overline{\chi}(a+k)$ $\exp(2\pi i\frac{a}{q}\frac{n}{k})$ ,$b(n,\chi)$ $=$ $\tau(\chi)-1$d(n) $\chi$ (n)e21$in/q,$$\tau(\chi)$ $d(n)$:241Proposition 1 Suppose that $T\geq 10$ and that $X$ satisfies $AqT<X<A’qT$ for any fixed$0<A<A’$ If $\chi$ is a primitive character of modulus $q$ , then we have$E(T, \chi)=\Sigma_{1}$ ,$\chi(T, X)+\Sigma_{2,\chi}(T, l(T, X/q))+R(T,\chi)$ ,where$\Sigma_{1,\chi}(T, y)$ $=$ $q^{3/4}( \frac{2T}{\pi})^{1/4}\sum_{n\leq y}\frac{|\overline{a(n,\overline{\chi})}|}{n^{3/4}}$$\mathrm{x}e(T, n/q)\cos(f(T,n/q)-\pi n/q+\arg\overline{a(n,\overline{\chi}})$),$\Sigma$2,$\chi(T, y)$ $=$ -2q1/2 $\sum_{n\leq qy}\frac{|\overline{b(n,\overline{\chi})}|}{n^{1/2}}(1$og $\frac{Tq}{2\pi n})^{-1}\mathrm{c}\mathrm{s}(g(T,n/q)+$ arg $\overline{b(n,\overline{\chi})}$)and $R(T,\chi)$ is $a$ oertain quantity which satisfies$R(T \chi|)\ll q(\log T)(\log qT)+\frac{q^{3/2+\epsilon}}{\log T}$ .$E$ (T) $E(T,\chi)$Theorem 1$f$ (t) $t$ $|f(t)|\leq c_{1}t^{1/4}$$\Lambda^{*}(t):=\frac{1}{\sqrt{2qt}}(\Lambda(2\pi t^{2}/q)+f(2\pi t^{2}/q))$$R_{\tau\mu}’(|u):=(1-|u|)(1+ \tau\sin(4\pi\theta\frac{\sqrt{\mu}}{q}u))$for $|u|\leq 1$ $\tau=1$ -1 $\theta>1$$\mu\in \mathrm{N}$Lemma 1 Let $\alpha(n)=a(n,\overline{\chi j})-a(n,\overline{\chi k})$ and no be $a$ minimum of numkrs $n$ such that$\alpha(n)\neq 0$ for fixed the pa$ir$ of characters. Suppose that $no<t^{3/2}$ . For every larye $t$ , wehave$\int_{-1}^{1}\Lambda^{*}(t+\theta u)K_{\tau,n_{0}}(u)du=$$\frac{|\overline{\alpha(n_{0})}|}{n_{0}^{3/4}}\{-\frac{\tau}{2}\sin(\frac{4\pi n_{0}^{1/2}}{q}t-\frac{\pi}{4}-\frac{\pi n_{\mathrm{O}}}{q}-\arg\alpha(n_{0}))+E\}$ (2)where $E$ is the error term estimated as$\ll$$\frac{\theta^{2}n_{0}^{3}}{q^{2}t^{4}}+\frac{q^{2}}{\theta^{2}n_{0}}(1+O(\frac{n_{\mathrm{O}}}{t^{2}}))^{-2}+\frac{n_{0}}{t^{2}}+\frac{n_{0}^{3/2}}{qt}$$\frac{n_{0}^{3/4}}{|\alpha(n_{0})|}\{\frac{1}{t^{3/8-e}}(\frac{\theta^{2}}{q}+\frac{q}{\theta}+n_{0}^{1/2})$$+ \frac{1}{t^{1/2-\epsilon}}(\frac{q^{1/2}}{\theta}+\frac{n_{0}^{1/2}}{q^{1/2}}+\frac{q}{t}+\frac{\theta}{q^{1/2}}+\theta+q^{1+\epsilon})$$+ \frac{q^{2}}{\theta^{2}}\sum_{nn\leq t^{3/2},\neq n_{0}}\frac{|\alpha(n)|}{n^{7/4}}\{1+(|1-\sqrt{\frac{n_{0}}{n}}|+O(\frac{n}{t^{2}}))^{-2}\}+\frac{c_{1}}{q^{3/4}}\}$ .242Remark 2 In [5] we proved that the number no exists for any fixed primitive characterswith $\chi j\neq\chi$k. However we do not know the upper bound of $n_{0}$ and a lower bound of$|\alpha(n_{\mathrm{O}})|$ with respect to $q$ . Thus we could not clarify the dependency of three number $T_{\mathit{0}\}}$$c_{1}$ and $c_{2}$ .Lemma 1 Theorem 1$q$ $n_{0}$ $c_{1}$$\theta$ $t$ $E$ ’(2) $\tau=1$ $\tau=-1$$\Lambda^{*}(w)$( [6] )(i) (Proposition y(Lemma 1)$(\mathrm{i}\mathrm{i}\mathrm{i})\mathrm{L}\mathrm{e}\mathrm{m}\mathrm{m}\mathrm{a}1$ $\Lambda(T)$(ii) $K_{\tau,\mu}(u)$ $\Lambda(T)$$|_{1}|$$(\mathrm{i}\mathrm{i})(\mathrm{i}\mathrm{i}\mathrm{i})$ D.R.Heath-Brown and K.Tsang [4]$E$(T)$E$(T)( $q=1$ $I\mathrm{f}_{\tau,1}$ (u)) $E$ (T) $|_{1}$(ii)$K_{\tau,\mu}(u)$Riemann zeta function and sums of Dirichlet $L$ functions$\zeta(s)$ $0<\Re s<1$4 Review of Atkinson’ $\mathrm{s}$ formula for the mean square of the$|^{1}$$I_{\sigma}(T)= \int_{0}^{T}|\zeta(\dot{\sigma}+ti)|^{2}d$t$I_{1/2}$ (T) $I$ (T) $E(T)=$$I(T)-T\log T-T(2\gamma-1-\log 2\pi)$ $E$(T)$E$(T) $\zeta(1/2+it)$$E$ (T) $|\zeta(1/2+it)|$ $|$$E$ (T)$|||$$|$2431949 F.V.Atkinson [1] $E$ (T) ():$E(T)$ $=$ $( \frac{2T}{\pi}$) $1/4 \sum_{n\leq X}(-1)^{n}\frac{d(n)}{n^{3/4}}e$(T, $n$) $\cos(f(T, n))$-2 $\sum$ $\frac{d(n)}{n^{1/2}}(\log\frac{T}{2\pi n})^{-1}\cos(g(T, n))+O(\log T)$, (3)$n\leq l(T,X)$$T\ll X\ll T$30 1978 Heath-Brown [3]:$\int_{2}^{T}E(t)^{2}dt=\frac{2}{3}\frac{1}{\sqrt{2\pi}}\sum_{n=1}^{\infty}\frac{d(n)^{2}}{n^{3/2}}T3/2+$F(T)(4)$F$(T) $F(T)=O(T^{5/4}\log 2T)$ () (3)T(4) A.Good $E(T)=\Omega(T^{1/4})$Heath-Brown (3)M.Jutila [9]$E$ (T)A.Ivic’ $E(T)\ll T^{3\mathrm{S}/108+}$ ‘ [7]Section 15.5 1989 J.LHafner and A.Ivic’[2]:$\int_{2}^{T}E(t)dt$ $=$ $\pi T+\frac{1}{2}(\frac{2T}{\pi})^{3/4}\sum_{n=1}^{\infty}(-1)^{n}\frac{d(n)}{n^{5/4}}\sin(2\sqrt{2\pi nT}-\frac{\pi}{4})$$+O(T^{2/3}\log T)$ , (5)$E(T)=\Omega_{+}((T\log T)^{1/4}(\log\log T)^{(3+1\text{ }\mathrm{g}4)/4}\exp(-c\sqrt{\log\log 1o\mathrm{g}T}))$,$E(T)= \Omega_{-}(T^{1/4}\exp(c\frac{(1\mathrm{o}\mathrm{g}1\mathrm{o}\mathrm{g}T)^{1/4}}{(1\mathrm{o}\mathrm{g}1\mathrm{o}\mathrm{g}1\mathrm{o}\mathrm{g}T)^{3/4}}))$ ,$E(T)\ll T^{139/429}($log $T)^{1467/429}$ , $c$ (3) ()(3) $E$(T)$\overline{E}_{\sigma}(T)$ $-I_{\sigma}(T)$ 1990 K.Matsumoto [13]$E_{\sigma}(T)$ $1/2<\sigma<3/4$ $E_{\sigma}(T)$(4) $\sigma=3/4$ $E_{\sigma}(T)$$\text{ }$ K.Matsumoto and T.Meurman [15] [16] $3/4\leq\sigma<1$Remark 34 [$lf$ Y.Motohas$fB\mathit{0}$]244$L$ (s, $\chi$) $\sum_{\chi}$ $\mathrm{m}\mathrm{o}\mathrm{d} q$$I_{\sigma}(T, q)= \int_{0}^{T}\sum_{\chi}|$L $(\sigma+it, \chi)$ $|^{2}$dt.$L$ (s, $\chi$) 1986 T. Meurman [17] $I_{1/2}$ (T, $q$ ) ( $I$ (T, $q$) )1993 A. $\mathrm{L}\mathrm{a}\mathrm{u}\mathrm{r}\mathrm{i}\mathrm{n}\check{\mathrm{c}}\mathrm{i}\mathrm{k}\mathrm{a}\mathrm{s}$ [12] MeurmanH. Nakaya[21][22] $1/2<\sigma<1$ ,[13], [15], [16]$L$ (s, $\chi$) ?$\zeta(s)$?$I$ (T)$I$ (T, $q$) $\sum_{\chi}$$I$ (T)$L$ $L$$\sum_{\chi}L$ (s, $\chi$) $\mathrm{L}\mathrm{b}$ ()$I$ (T, $\chi$)T5 Proposition 1$L$ (u, $\chi$) $L(v, \overline{\chi})$ $\Re u>1$ and $\Re v>1$:$L(u,\chi)$L(v, $\overline{\chi}$) $=( \sum_{m=n}+\sum_{m<n}+\sum_{m>n})\frac{\chi(m)}{m^{u}}\overline{\frac{\chi(n)}{n^{v}}}$ . (6)$m=n$$L(u+v,\chi_{0})$ $L$$L$ $varrow 1-u$$u=1/2+it$ $\int_{0}^{T}\ldots d$t$E$ (T)Y.Motohashi [19] AtkinsonAtkinsonMotohashi$L$(6)$varrow 1-u$ :(Motohashi’sLe$mma$ 1of $f\mathit{1}\mathit{9}i$) If $0<\Re u<1$ , then$L(u,\chi)$L(1-u, $\overline{\chi}$) $=$$\frac{\phi(q)}{q}\{\frac{1}{2}(\frac{\Gamma’(1-u)}{\Gamma(1-u)}+\frac{\Gamma^{l}(u)}{\Gamma(u)})+(2\gamma-\log\frac{2\pi}{q}+2\sum_{p1q}\frac{1\mathrm{o}\mathrm{g}p}{p-1})$}$+g(u, \chi)+$ g(1-u, $\overline{\chi}$), $(7)$245where $g(u, \chi)$ is the analytic continuation of$g(u, \chi)=\sum_{n=1}^{\infty}a(n, \chi)$h $(u, n)+ \sum_{n=1}^{\infty}\overline{a(n,\overline{\chi}})h$( $u,$ -n), (8)and where$h(u, x)= \int_{0}^{\infty}\frac{e^{-2\pi ixy/q}}{y^{u}(1+y)^{1-u}}dy$.The series (8) is convergent when $Reu<0$ .[19] (7) $L$ (s, $\chi$)(7)’()(7) $\sum_{n\leq x}a$ (n, $\chi$) $g$ (u, $\chi$)$\int_{1/2}^{1/2+1T}.L(u)L(1-u)d$u($\int_{a}^{b}g$ (x)e|.f$(x)d$x )Atkinson [1]$I$ (T) lemma $I$ (T, $q$)$I$ (T, $\chi$)$\int_{0}^{\infty}\frac{\exp\pm(i2\pi k\pm it1\mathrm{o}\mathrm{g}\frac{1+y}{y})}{y^{1/2}(1+y)^{1/2}1\mathrm{o}\mathrm{g}\underline{1}_{A}+\mathrm{y}}$dy(9)$t=0$ $\ulcorner\zeta(s)$$\sum_{\chi}L$ (s, $\chi$ ) $t=0$$I$ (T) $I$ (T, $q$)$I$ (T) 2 $\int_{0}^{T}|\zeta(\ldots)|^{2}dt=\int_{-T}^{T}|\zeta(\ldots)|^{2}d$ t() (9) $I$ (T, $q$)(9) $I(T,\chi)$$\int_{0}^{T}|L(1/2+it, \chi)$ |2dt($\int_{T/2}^{T}+\int_{T/2^{2}}^{T/2}+\ldots+\int_{T/2^{L}}^{T/2}$”$1+ \int_{0}$Tl2’) $|$L$(1/2 +it, \chi)|^{2}$dt, (10)with $L=$ [($\log T$ -log log $T)/\log 2$]$L(1/2+it, \chi)\ll(qt)^{1/4+\epsilon}$ ,lemmaHafner and $\mathrm{I}\mathrm{v}\mathrm{i}\acute{\mathrm{c}}$ [2](5) $\int_{2}^{T}E$ (t)dt :($\int_{T/2}^{T}+7T/2^{2}T/2+...+\int_{T/2^{L}}^{T/2^{L-1}}+\mathit{1}T/2^{L}$) $E(t)dt$ .248$E$(t)lemmaA $\mathrm{B}$$T$Jutila [10]( )(5)(10)$K$ (t, $n$) $\tilde{K}$ (t, $n$):K( $n$) $=$ $| \overline{a(n,\overline{\chi})}|\frac{1}{\sqrt{2}}\frac{1}{\sqrt{n/q}}\cos(f$( $n/q$) $- \pi\frac{n}{q}+\arg\overline{a(n,\overline{\chi})})$$\{$$\tilde{K}(t, n)=2q^{1}$$\sum_{\eta_{2}<n\leq\eta 1}$$\eta_{2}=ql(2T,$ $q^{-1}2X$Jutila[10]$K$ (2T, $n$ )$( \frac{t}{2\pi n/q}+\frac{1}{4})^{1/4}\mathrm{a}\mathrm{r}\sinh\sqrt{\frac{\pi n}{2tq}}\}^{-1}$:$\tilde{K}(2T, n)\sim-\sum_{X<n\leq 2X}K(2T, n)$$(T\prec\infty)$ , (11)$/2_{\frac{|\overline{b(n,\overline{\chi})}|}{n^{1/2}}}( \log\frac{tq}{2\pi n})^{-1}\cos(g(t, n/q)+\arg\overline{b(n,\overline{\chi})})$ .$|^{1}$$)$ $\eta_{1}=ql$ (2T, $q^{-1}X$) Hiner and $\mathrm{I}\mathrm{v}\mathrm{i}\acute{\mathrm{c}}$$\tilde{K}$ (2L\rightarrow$\sum_{n<\pi}\overline{b(n,\overline{\chi})}$ VoronoiVoronoi $a$ (n, $\chi$)$b$ (n, $\chi$) $\sum_{n<x}a$ (n, $\chi$) Voronoi$b$(n, $\chi$ ) $b$ (n, $\chi$) Voronoi$a$ (n, $\chi$) ?Jutila $b$ (n, $\chi$)( )Jutila$1^{1}|’|$(11)$\ulcorner K$ (2T, $n$) $\tilde{K}$ (2T, $n$) $K$ (2T, $n$ ) $\tilde{K}$ (2T, $n$) 1$T$$K$ (2T, $n$ ) A A$P$(T) $P(T)$$A+P$(T)( [5] )$P$ (T) Jutla [10]( $b$(n, $\chi$) Voronoi )T.Meurman $I$ (T, $\chi$) [18]$\gamma_{\mathrm{J}}\text{ _{ },I(T,\chi)}$$|^{1}$247Remark 4 f-TT|L(l/2+i \chi )|2d(9) $I$ (T) $I$ (L $q$)6$\Lambda(T)$ $L(s, \chi)^{2}$$E$ (T)$\Lambda(T)$better$L$ (s, $\chi$)\Lambda (T)=Oq(Tl/3+e)( $\epsilon$ )$|L(1/2+it,\chi j\rangle$ $-L$ ( $1/2+it,\chi$k)|() $L$References[1] $\mathrm{F}.\mathrm{V}$ . Atkinson, The mean-value of the Riemann zeta function, Acta Math. 81 (1949),353-376.[2] $\mathrm{J}.\mathrm{L}$ . Hafner and A. Ivic’, On the mean-square of the Riemann zeta function on thecritical line, J. Number Theory 32 (1989), 151-191.[3] $\mathrm{D}.\mathrm{R}$ . Heath-Brown, The mwan value of theorem for the Riemann zeta-function, Math-ematika 25 (1978), 177-184.[4] $\mathrm{D}.\mathrm{R}$ . Heath-Brown and K. Tsang, Sign changes of $E(T),$ $\Delta(X)$ and $P(X)$ , J. NumberTheory 49 (1994), 73-83.[5] H. Ishikawa, The rreean square of Dirichlet $L$ functions and a difference betrryeen thevalues of $L(1/2+it,$ $\chi j)$ and $L(1/2+it, \chi_{k})$ , submitted.[6] H. Ishikawa, The rnean square of Dirichlet $L$ functions and a difference beteryeen thevalues of $L(1/2+it,Xj)$ and $L(1/2+it,$ $\chi_{k})II$, submitted.[7] A. Ivic’, The Riemann Zeta Function, Wiley-Interscience Publication, New York, 1985.[8] A. $\mathrm{I}\mathrm{v}\mathrm{i}\acute{\mathrm{c}}$ and H. J. J. te Riele, On the zero of the error term for the mean square of$|\zeta$ ($1/2+i$t)l Math. Comput. 56(1991), 303-328.[9] M. Jutila, Riemann’s zeta function and the divisor problem, Ark. Math. 21 (1983),75-96.248[10] M. Jutila, Transformation formula for Dirichlet polynomials, J. Number Theory18(1984), 135-156.[11] M. Katsurada and K. Matsumoto, A weighted integral approach to the mean squreof Dirichlet $L$-functions, in Ntrmber Theory and its Applications”, K. Gy\"ory andS. Kanemitsu(eds.), Developments in Math. Vol. 2, Kluwer Acad. Publ., 1999, pp.19&229[12] A. $\mathrm{L}\mathrm{a}\mathrm{u}\mathrm{r}\mathrm{i}\mathrm{n}\check{\mathrm{c}}\mathrm{i}\mathrm{k}\mathrm{a}\mathrm{e}$ , The Atkinson formula for $L$-functions near the critical line, Liet.Math. Rink. 33(1993)435-454(in Russian) $=\mathrm{L}\mathrm{i}\mathrm{t}\mathrm{h}\mathrm{u}\mathrm{a}\mathrm{n}\mathrm{i}\mathrm{a}\mathrm{n}$ Math. J. 35 (1993) 337-351.[13] K.Matsumoto, The mean square of the Riemann zeta-function in the critical strip,Japanese J. Math. 15(1989), $1rightarrow 13$ .[14] K.Matsumoto, Recent Developments in the Mean Square Theory of the RiemannZeta and other Zeta-Functions, in $u$Number Theory” (R.P.Bambah, $\mathrm{V}.\mathrm{C}$ . Dumir and$\mathrm{R}.\mathrm{J}$ . Hams-Gill $\mathrm{e}\mathrm{d}\mathrm{s}.$ ) Hindustan Book Agency and Indian National Science Acad. dis-tributed (outside India) by Birkh\"auser, 2000, 241-286.[15] K.Matsumoto and T.Meurman, The mean square of the Riemann zeta-function inthe critical strip $\mathrm{I}\mathrm{I}$ , Acta Arith. 68 (1994), 369-382.[16] K.Matsumoto and T.Meurman, The mean square of the Riemann zeta-function i$\mathrm{n}$the critical strip III, Acta Arith. 64 (1993), 357-382.[17] T.Meurman, A generalization of Atkinson’s formula to $L$-functions, Acta Arith.47 (1986), 351-370.[18] T.Meurman, Identities of the Atkinson type for $L$-functions, unpublished manuscript.[19] Y.Motohashi, A note on the mean value of the zeta and $L$-functions $\mathrm{I}\mathrm{I}$ , Proc. JapanAcad. 61 $\mathrm{A}(1985)$ , 313-316.[20] Y.Motohashi, , 21 21,(1999).[21] H.Nakaya, The mean squre of the Dirichlet $L$-function in the critical strip, Liet. Math.Rink. 40(2000), 201-213.[22] H.Nakaya, On the mean square formula for Dirichlet Dfunctions, ${\rm Res}$ . Proc. TokyoMetrop. College of Aeronautical Engeineering. 38 (2000), 1-8.

$L(1/2+it,\chi_j)$と$L(1/2+it,\chi_k)$の値の差について　(解析的整数論とその周辺)

http://repository.kulib.kyoto-u.ac.jp/dspace/bitstream/2433/25758/1/1384-35.pdf

$L(1/2+it,\chi_j)$ と $L(1/2+it,\chi_k)$ の値の差について　(解析的整数論とその周辺)

Abstract

Similar works

Full text

Available Versions

Kyoto University Research Information Repository

L(1/2+it,χj)L(1/2+it,\chi_j)L(1/2+it,χj​)とL(1/2+it,χk)L(1/2+it,\chi_k)L(1/2+it,χk​)の値の差について (解析的整数論とその周辺)

Abstract

Similar works

Full text

Available Versions

Kyoto University Research Information Repository

$L(1/2+it,\chi_j)$ と $L(1/2+it,\chi_k)$ の値の差について　(解析的整数論とその周辺)