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1 (Kohji Matsumoto) 1 [18] 1999 $- \mathrm{b}^{\backslash }$ $\zeta(s \alpha)$ Hurwitz $\Re s>1$ $\Sigma_{n=0}^{\infty}(\alpha+n)^{-S}$ $\zeta_{1}(s \alpha)=\zeta(s \alpha)-\alpha^{-}s$ $\alpha$ $\alpha$ $\int_{0}^{1} \zeta_{1}(s \alpha) ^{2}d\alpha$ (1) [16] 1994 Barnes (1) $\zeta(u \alpha)\zeta(v \alpha)$ $=$ $( \sum_{m=n}+\sum_{m<n}+\sum_{>mn}\mathrm{i}(\alpha+m)^{-}u(\alpha+n)^{-v}$ $=$ $\zeta(u+v \alpha)+f(u v;\alpha)+f(v u;\alpha)$ (2) (Atkinson ) $f(u v; \alpha)=\sum_{m=0}^{\infty}(\alpha+m)^{-}u\sum_{n=1}\infty(\alpha+m+n)-v$ (3) $f(u v;\alpha)$ [4]

2 260 Barnes $\zeta_{2}(v;\alpha w)=\sum m\infty=0\sum_{n=0}(\alpha+m+nw)^{-v}\infty$ (4) ( $\alpha>0$ $w>0$ ; notation ) (3) $f(u v;\alpha)$ $\tilde{\zeta}_{2}(u v;\alpha w)=\sum^{\infty}(m=0\alpha+m)^{-}u\sum_{n=0}(\alpha+m+nw)^{-v}\infty$ (5) (3) (4) $f(u v;\alpha)$ $\tilde{\zeta}_{2}(u v;\alpha w)$ $u=0$ Barnes Barnes Barnes $\zeta_{2}(v;\alpha w)$ $w$ ( $warrow\infty$ ) Corollary 7 check : (E1) $\zeta_{2}(v\cdot\alpha w)$ -? (E2) - $\tilde{\zeta}_{2}(u v;\alpha w)$ $\sum_{m}\sum_{n}(\alpha+m+nw1)-u(\beta+m+nw_{2})-v$ (6) $u=v$ (5) Shintani degenerate Shintani (6)? contour integral $\tilde{\zeta}_{2}(u v;\alpha w)$ $u=0$ $u$ $0$ (6) contour integral Eisenstein $G_{v}(w)= \sum_{m}\sum_{n}(m+nw)^{-v}$ (7)

3 $\alpha$ 261 $\mathrm{h}$ $w$ $m=n=0$ (7) Barnes (4) (7) $w$ ( ) [17] $w$ $\Re w>0$ $w$ $G_{v}(w)$ $\Re w>0$ (7) $m$ $n$ Riemann $0$ $\zeta(s)$ $\sum_{m=1n}^{\infty}\sum_{=1}\infty(m+nw)^{-v}+\sum_{m=1n}\sum_{=1}\infty\infty(m-nw)^{-v}$ $+ \sum_{m=1}^{\infty}\sum_{n=1}\infty(-(m-nw))-v+\sum_{=m1}^{\infty}\sum_{1n=}\infty(-(m+nw))^{-v}$ (8) $\Re w>0$ [17] $\Re(-w)<0$ contour integral Barnes 1994 Mellin-Barnes (1) Dirichlet $\Sigma_{\chi} L(s \chi) 2$ contour integral $\langle$ contour integral Mellin-Barnes ([12] [17]). [17] [19] Hurwitz (1) Mellin-Barnes Journ\ ees Arithm\ etiques $([11])_{0}$ [17] Barnes $\sum_{m=0}^{\infty}\sum_{n=0}^{\infty}(\alpha+m)^{-}u(\alpha+\beta+m+n)^{-v}$ (9) ( ) 1996 Lerch $\Re u>1$ $\Re v>1$ (9) $\beta$

4 262 Mellin-Barnes Mellin-Barnes [13] [17] 1995 $*^{\backslash }$ (Universality Rankin-Selberg series ) o Hurwitz Mellin-Barnes [17] Mellin-Barnes [11] [17] [17] Barnes Shintani $p$ Zagier Dedekind $\mathrm{h}\mathrm{p}$ [7] Barnes (4) 20 Barnes (4) $\sum_{m_{1}=0\gamma}^{\infty}\cdots\sum_{m}\infty=0(\alpha+m1w1+m_{2}w2+\cdots+m_{rr}w)-v$ (10) ( $r$ ) $\alpha>0$ $w_{j}$ $w_{j}>0(1\leq i\leq r)$ Barnes $v$ Barnes Hardy Littlewood Dirichlet \epsilon - $*$ 70 Hecke (Shintani ) Barnes

5 Zagier [22] $\tilde{\zeta}_{r}(v_{1\ldotsr}v)=1\leq m1<m_{2}<\cdot<r\sum_{m}..m_{1}-v_{1}m_{2}-v2\ldots m_{r}-v_{r}$ (11) Euler Euler- Zagier sum Zagier $v_{j}$ $(\geq 2)$ ( ) ( [10] ) Euler-Zagier sum (11) $v_{r}$ [5] $r=2$ 1949 Atkinson [6] (11) (3) $\alpha=1$ Atkinson Riemann Poisson $\tilde{\zeta}_{r}(v_{1} \ldots v_{r})$ $r$ Euler-Maclaurin ([1]) Zhao [23] [5] $r$ $\tilde{\zeta}_{r}(v_{1} \ldots v_{r})$ [3] $(_{r}(v_{1} \ldots v_{r})\sim$ (11) $r$ Dirichlet ([2]) Hurwitz $\sum_{m_{1}=1}^{\infty}\cdots\sum_{r}(\alpha 1+m1)-v_{1}(\alpha 2+mm=\infty 11+m2)^{-v2}$ $\cross\cdots\cross(\alpha_{r}+m_{1}+m_{2}+\cdots+m_{r})-v_{r}$ (12) ( $1\leq i\leq r$ $0\leq\alpha_{j}<1$ ) [1] (12) [91 (12) $r=2$ Hurwitz [15] contour integral [17] Mellin-Barnes

6 264 Mellin-Blnes Mellin-Barnes $\Gamma(z)(1+\eta)^{-z}=\frac{1}{2\pi i}\int_{(c)}\mathrm{r}(\mathcal{z}+s)\gamma(-s)\eta sds$ (13) $z$ $\eta$ $\Re z>0$ $ \arg\eta <\pi$ $\eta\neq 0$ $c-i\infty$ $c+i\infty$ (13) $z=v$ $\eta=nw/(\alpha+m)$ $\Gamma(v)(\alpha+m)^{-u-v}$ $( \alpha+m)^{-}u(\alpha+m+nw)^{-v}=\frac{1}{2\pi i}\int_{c)}(\frac{\gamma(v+s)\gamma(-s)}{\gamma(v)}(\alpha+m)-u-v-s(nw)^{s_{d_{s}}}$ (14) $\Re u>1$ $\Re v>1$ $-\Re v<c<-1$ $m$ $n$ $\tilde{\zeta}_{2}(u v;\alpha w)=\zeta(u+v \alpha)$ $+ \frac{1}{2\pi i}\int_{()}c\frac{\gamma(v+s)\gamma(-s)}{\gamma(v)}\zeta(u+v+s \alpha)\zeta(-s)w^{s}ds$ (15) $s=1-u-v$ $s=-v-n$ ( $n=012$ $\ldots$ ) shift $w$ shift Mellin-Barnes [17]. (15) $s=-1012$ $\ldots$ shift? Mellin-Barnes shift (15) shift $warrow \mathrm{o}$ contour integral shift $warrow \mathrm{o}$ state $N$ $warrow\infty$ $\tilde{\zeta}_{2}(u v;\alpha w)=\zeta(u+v \alpha)+\frac{\gamma(1-u)\mathrm{r}(u+v-1)}{\gamma(v)}\zeta(u+v-1)w^{1}-u-v$ $+ \sum_{n=0}^{n-}1\zeta(u-n \alpha)\zeta(v+n)w-v-n+o( w ^{-\Re N}v-)$ (16)

7 265 $\Re u<n+1$ $\Re v>-n+1$ ( ) $u$ $warrow \mathrm{o}$ $\tilde{\zeta}_{2}(u v;\alpha w)=\zeta(u+v \alpha)-\frac{1}{1-v}\zeta(u+v-1 \alpha)w^{-1}$ $+ \sum_{n=0}^{n-}1\zeta(u+v+n \alpha)\zeta(-n)w+no( w N)$ (17) $\Re v>-n$ $\Re(u+v)>1-N$ $u=0$ Barnes (16) [17] (17) $warrow \mathrm{o}$ $\eta$ (13) $ \arg\eta <\pi$ (15) $ \arg w <\pi$ $w\neq 0$ (16) (17) $\theta_{0}(0<\theta_{0}<\pi)$ $ \arg w \leq\theta_{0}$ Eisenstein (16) (17) Eisenstein (7) $w$ (16) (17) [21] Dedekind Dedekind 5 discussion Mellin-Barnes contour integral 7 (E2) $*^{\backslash }$ Mellin-Barnes Shintani (6) $\sum_{m=0}\sum_{n=0}(a+m+(b+n)w1)^{-}u(a+m+(b+n)w_{2})-v$ (18) Mellin-Barnes ([20]) $a>0$ $b>0$ $w_{1}$ $w_{2}$ ( $u=v$ $*^{\backslash }-F$ $[8]_{\text{ }})$ Shintani cf. Hida $w_{1}\leq w_{2}$ $w_{1}=w_{2}$ (18) Barnes $w_{1}<w_{2}$ (13) (15) $\eta=\frac{(b+n)(w2-w_{1})}{a+m+(b+n)w_{1}}$ (18) $\tilde{\zeta}_{2}(-s u+v+s;b b+aw_{1 1}^{-1-1}w)$ Mellin-Barnes $\tilde{\zeta}_{2}(u v;\alpha \beta w)=m\sum_{=0}^{\infty}(\alpha+m)^{-u}\sum_{n=0}^{\infty}(\beta+m+nw)^{-v}$ (19)

$266 (19) (5) (5) shift (19) Shintani (18) (16) (17) (19) (E2) Mellin-Barnes (E1) $u=0$ (14) (15) $u=0$ Barnes (5) Mellin-Barnes (5) (19) Shintani contour integral Shintani * - Shintani $warrow\infty$$

8 266 (19) (5) (5) shift (19) Shintani (18) (16) (17) (19) (E2) Mellin-Barnes (E1) $u=0$ (14) (15) $u=0$ Barnes (5) Mellin-Barnes (5) (19) Shintani contour integral Shintani * - Shintani $warrow\infty$ $warrow \mathrm{o}$ (16) (17) 10 TeX le ( ) 11 TeX le Euler-Zagier sum 7 19 $r=2$ $*^{\backslash }-$ Hurwitz ((12) $r=2$ ) (19) $w=1$ $r$ $-$ contour integral $r=2$ $\lceil_{r=2}$ $r=2$ $r$ $r=2$ ( ) $r\geq 3$ Barnes

9 267 Shintani - Mellin-Barnes (12) $r$ $r-1$ Mellin-Barnes $r=1$ Hurwitz $r$!( Vo11091 ) $\alpha_{j}>0$ $v_{j}$ $w_{j}$ $(1 \leq i\leq r)$ $\Re v_{j}>1$ $\tilde{\zeta}_{r}(v_{1} \ldots v_{r}; \alpha_{1} \ldots \alpha_{r};w1 \ldots w_{r})$ $= \sum_{=m_{1}0m}^{\infty}\cdots\sum_{r}^{\infty}=0(\alpha_{1}+m_{1}w1)-v1(\alpha_{2}+m_{1}w1+m_{2}w_{2})^{-v_{2}}$ $\cross\cdots\cross(\alpha_{r}+m1w1+m_{2}w2+\cdots+mrwr)-v_{r}$ (20) Barnes (10) Euler-Zagier (11)(12) (20) $w_{j}$ $w_{j}$ $w_{j}>0$ (20) $v_{1}=v_{2}=\cdots=vr-1=0$ Barnes $*^{\backslash }-p(10)$ ( ) Barnes (20) Mellin-Barnes $w_{1}=w_{2}=\cdots=wr=1$ $\tilde{\zeta}_{r}(v_{1} \ldots v_{r}; \alpha_{1} \ldots \alpha_{r})$ $= \sum_{=m_{1}0m}^{\infty}\cdots\sum_{r}^{\infty}(\alpha 1+m1)^{-}v1(\alpha 2+m_{1}+m_{2})^{-v2}=0$ $\mathrm{x}\cdots\cross(\alpha_{r}+m_{1}+m_{2}+\cdots+mr)^{-}v_{r}$ (21) $\alpha_{1}<\alpha_{2}<\cdots<\alpha_{r}$ (22) (21) (12) ((21) $m_{j}=0$ ) (22) $\mathrm{c}^{r}$ $\ldots$ $v_{r}$ (21) $v_{1}$

10 268 (13) $z=v_{r}$ $\eta=\frac{\alpha_{r}-\alpha_{r-1}+.m_{r}}{\alpha_{r-1}+m_{1}+m2++mr-1}.$. Mellin-Barnes $\tilde{\zeta}_{r}(v_{1} \ldots v_{r};\alpha_{1} \ldots \alpha r)$ $= \frac{1}{2\pi i}\int_{(c})\frac{\gamma(v_{r}+s)\mathrm{r}(-\mathit{8})}{\gamma(v_{r})}\zeta(-s \alpha_{r}-\alpha_{r-1})$ $\cross\tilde{\zeta}_{r-1}(v_{1\ldotsr}v-2 vr-1+v_{r}+s;\alpha_{1} \ldots \alpha_{r-1})ds$ (23) shift shift $r-1$ order [18] TeX le (16) (17) Eisenstein Shintani $r$ * $=ff$ Mellin-Barnes shift explicit Hurwitz Mellin-Barnes $k$ $q$ $\sum_{a=1}^{q} \zeta(s$ $\frac{a}{q}) ^{2k}$ (24) $\sum_{a=1}^{q}\tilde{\zeta}r(v_{1}$ $v_{2}$ $.$. $v_{r}; \frac{a}{q}$ $\frac{a}{q}+1$ $\frac{a}{q}+(r-1))$ $$ $\ldots$ (25) (25) $q$ Mellin-Barnes ( ) $k$ (24) $q$ $k=1$ ([14]) $k$ Mellin-Barnes (20) $w_{j}$ [2]

11 269 Mellin-Barnes [13] (1) - Shintani Mellin-Barnes Hecke [19] [1] S.Akiyama S.Egami and Y.Tanigawa An analytic continuation of multiple zeta functions and their values at non-positive integers preprint. [2] S.Akiyama and H.Ishikawa On analytic continuation of multiple -functions and related zeta-functions preprint. [3] S.Akiyama and Y.Tanigawa Multiple zeta values at non-positive integers preprint. $\sum_{n=1}^{\infty}\frac{\cot\pi n\alpha}{n^{s}}$ [4] T.Arakawa Dirichlet series Dedekind sums and Hecke -functions for real quadratic fields Comment. Math. Univ. St. Pauli 37 (1988) [5] T.Arakawa and M.Kaneko Multiple zeta values poly-bernoulli numbers and related zeta functions Nagoya Math. J. 153 (1999) [6] $\mathrm{f}.\mathrm{v}$.atkinson The mean-value of the Riemann zeta function Acta Math. 81 (1949) [7] 1998 (http: $//\mathrm{m}\mathrm{a}\mathrm{t}\mathrm{h}\mathrm{a}\mathrm{l}\mathrm{g}.\mathrm{g}\mathrm{e}$.niigata-u. $\mathrm{a}\mathrm{c}.\mathrm{j}\mathrm{p}/\mathrm{m}\mathrm{a}\mathrm{t}\mathrm{h}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{k}.\mathrm{h}\mathrm{t}\mathrm{m}$ ). [8] H.Hida Elementary Theory of -Functions and Eisenstein Series London Math. Soc. Student Texts 26 Cambridge Univ. Press [9] ) On asum related to amultiple -function $*^{\backslash }-$ [10] 1097 (1999) [11] M.Katsurada An application of Mellin-Barnes type integrals to the mean square of Lerch zeta-functions Collect. Math. 48 (1997)

12 270 [12] M.Katsurada An application of Mellin-Barnes type integrals to the mean square of -functions Liet. Mat. Rink. 38 (1998) [13] M.Katsurada Power series and asymptotic series associated with the Lerch zetafunction Proc. Japan Acad. (1998) $74\mathrm{A}$ [14] M.Katsurada and K.Matsumoto Discrete mean values of Hurwitz zeta-functions $69\mathrm{A}$ Proc. Japan Acad. (1993) [15] M.Katsurada and K.Matsumoto Explicit formulas and asymptotic expansions for certain mean square of Hurwitz zeta-functions I Math. Scand. 78 (1996) [16] Hurwitz 1060 (1998) [17] K.Matsumoto Asymptotic series for double zeta double gamma and Hecke L- functions Math. Proc. Cambridge Phil. Soc. 123 (1998) [18] K.Matsumoto Asymptotic expansions of double zeta-functions of Barnes of Shintani and Eisenstein series preprint. [19] K.Matsumoto Asymptotic series for double zeta double gamma and Hecke L- functions II preprint. [20] T.Shintani On a Kronecker limit formula for real quadratic fields J. Fac. Sci. Univ. Tokyo Sect. IA Math. 24 (1977) [21] T.Shintani A proof of the classical Kronecker limit formula Tokyo J. Math. 3 (1980) [22] D.Zagier Values of zeta functions and their applications in First European Congress of Mathematics Vol.II Invited Lectures (Part 2) A.Joseph et al.(eds.) Progress in Math. 120 Birkh\"auser 1994 pp [23] J.Zhao Analytic continuation of multiple zeta functions Proc. Amer. Math. Soc. to appear.

(Masatake MORI) 1., $I= \int_{-1}^{1}\frac{dx}{\mathrm{r}_{2-x})(1-\mathcal{i}1}.$ (1.1) $\overline{(2-x)(1-\mathcal{i})^{1}/4(1

$(Masatake MORI) 1., $I= \int_{-1}^{1}\frac{dx}{\mathrm{r}_{2-x})(1-\mathcal{i}1}.$ (1.1) $\overline{(2-x)(1-\mathcal{i})^{1}/4(1$ 1040 1998 143-153 143 (Masatake MORI) 1 $I= \int_{-1}^{1}\frac{dx}{\mathrm{r}_{2-x})(1-\mathcal{i}1}$ (11) $\overline{(2-x)(1-\mathcal{i})^{1}/4(1+x)3/4}$ 1974 [31 8 10 11] $I= \int_{a}^{b}f(\mathcal{i})d_{x}$