数理解析研究所講究録
第 1874 巻 2014 年 183-189

183

The asymptotic behavior of multiple zeta functions at
non-positive integers
TOMOKAZU ONOZUKA

1Introduction

The Euler-Zagier multiple zeta function$\zeta_{d}(s_{1}, \cdots, s_{d})$is defined by$\zeta_{d}(s_{1}, \cdots, s_{d}) :=\sum_{m1^{=1}m}^{\infty}\cdots\sum_{d^{=1}}^{\infty}\frac{1}{m_{1}^{s_{1}}(m_{1}+m_{2})^{s_{2}}\cdots(m_{1}+\cdots+m_{d})^{s}d}$(1.1)

where $s_{i}(i=1, \cdots, d)$ are complex variables. Matsumoto [4] proved that the series (1.1)
is absolutely convergent in$\{(s_{1}, \cdots, s_{d})\in \mathbb{C}^{d}|\Re(s_{d}(d-k+1))>k(k=1, \cdots, d)\}$where $s_{d}(n)=s_{n}+s_{n+1}+\cdots+s_{d}(n=1, \cdots, d)$ . Akiyama, Egami and Tanigawa [1] andZhao [7] proved the meromorphic continuation to the whole space independently.
The function
has singularities on$\zeta_{d}(s_{1}, \cdots, s_{d})$(1.2)$\{\begin{array}{l}s_{d}=1,s_{d-1}+s_{d}=2,1,0, -2, -4, \cdots,s_{d}(d-j+1)\in \mathbb{Z}_{\leq j}(j=3,4, \cdots, d) ,\end{array}$where$\mathbb{Z}_{\leq j}$$(-r_{1}, \cdots.

is the set of integers less than or equal tois defined similarly. Therefore
lies on the set of singularities. Moreover, it is an indeterminacy of. For example, Sasaki [6] proved that$j;\mathbb{Z}_{\geq j}$-r_{d})\in \mathbb{Z}_{\leq 0}^{d}$$\zeta_{d}(s_{1}, \cdots, s_{d})$$s_{3} arrow 0_{S2}\lim hm\lim_{arrow 0_{s1}arrow 0}\zeta_{3}(s_{1}, s_{2}, s_{3})=-\frac{3}{8}$

,

(1.3)

$S arrow 0hm\lim_{arrow 0}\lim_{3}\zeta_{3}(s_{1}, s_{2}, s_{3})=-\frac{1}{4}$

.

(1.4)

Since $(0,0,0)$ is an indeterminacy of$(1.3)$ and (1.4) give different values.
Akiyama, Egami and Tanigawa [1] defined the regular values by$\zeta_{3}(s_{1}, s_{2}, s_{3}),$$\zeta_{d}(-r_{1}, \cdots, -r_{d}):=\lim_{1d}\cdots\lim_{dsarrow-r1sarrow-r}\zeta_{d}(s_{1}, \cdots, s_{d})$,184
and Akiyama and Tanigawa [2] considered the reverse and central values given by,$\zeta_{d}^{R}(-r_{1}, \cdots, -r_{d}):=\lim_{dsdarrow-r}\cdots\lim_{s1^{arrow-r1}}\zeta_{d}(s_{1}, \cdots, s_{d})$,$\zeta_{d}^{c}(-r_{1}, \cdots, -r_{d}):=\lim_{\epsilonarrow 0}\zeta_{d}(-r_{1}+\epsilon, \cdots, -r_{d}+\epsilon)$

respectively. Further, Sasaki [6] generalized the regular and reverse values. He defined
multiple zeta values for coordinatewise limits by$\zeta_{d}(-r_{1}, \cdots, -r_{d})i_{1}i_{d}:=\lim_{j^{=d}}\cdots\lim_{jsarrow-rSj_{i_{j}=1}^{arrow-r}}\zeta_{d}(s_{1}, \cdots, s_{d})$,where $\{i_{1}, \cdots, i_{d}\}=\{1, \cdots, d\}$ . He obtained all multiple zeta values of depth 3 for coordinatewise limits. On the other hand, Komori [3] considered more general multiple zetafunctions, and he obtained multiple zeta values at non-positive integers given by$\zeta_{d}(-r)= \lim_{-,-r_{w}}\lim_{(z_{w(d)}d)}\zeta_{d}(z_{1}, \cdots, z_{d})w\ldots,$$\zeta_{d}(-r)=\zeta_{d}(-r_{1}, \cdots, -r_{d})= \lim_{\delta,\theta\theta_{1}\theta_{d}arrow 0}\zeta_{d}(-r_{1}+\delta\theta_{1}, \cdots, -r_{d}+\delta\theta_{d})$,. To obtain these
and
where
numbers.
multiple
Bernoullivalues by Komori’s method, we need to compute generalized
In the present paper, we calculate the asymptotic behavior of multiple zeta functions
at non-positive integers. By using that result, we can evaluate the limit values of multiple
zeta functions at non-positive integers. For example,$-r=(-r_{1}, \cdots, -r_{d})\in \mathbb{Z}_{\leq 0}^{d},$$w\in \mathfrak{S}_{d}$$\theta=(\theta_{1}, \cdots, \theta_{d})\in \mathbb{C}^{d}$$\lim_{\epsilonarrow 0}\zeta_{3}(\epsilon^{2}, \epsilon, \epsilon)=-\frac{1}{3}$

.

(1.5)

This limit value is not contained in the above 2 kinds of values, however by the result, we
can compute this value.

2

Main Theorem

In this section, we state the main theorem.
be the mth Bernoulli number, and $B(x, y)$ be the beta function. ForLetand, let $m_{d}(n),$andrespectively.
In$p_{n}+p_{n+1}+\cdots+p_{d}$and$m_{n}+m_{n+1}+\cdots+m_{d},$be$(a)_{n}:=\Gamma(a+n)/\Gamma(a)$.
is defined byaddition, the Pochhammer symbol$B_{rn}$$(m_{1}, \cdots, m_{d})\in \mathbb{Z}_{\geq 0}^{d},$$(p_{1}, \cdots,p_{d})\in \mathbb{Z}_{\geq 0}^{d}$$p_{d}(n)$$(\epsilon_{1}, \cdots , \epsilon_{d})\in \mathbb{C}^{d}$$\epsilon_{d}(n)$$\epsilon_{n}+\epsilon_{n+1}+\cdots+\epsilon_{d}$$(a)_{n}$$\epsilon_{d}(j)\neq 0(j=1, \cdots, d),$Theorem 1. Suppose thatas $(\epsilon_{1}, \cdots, \epsilon_{d})arrow(0, \cdots, 0)(j=1, \cdots, d, k=j, \cdots, d)$ . Then forand$\epsilon_{j}\neq 0,$$|\epsilon_{k}/\epsilon_{d}(j)|\ll 1$$| \epsilon_{1}|+\cdots+|\epsilon_{d}|\leq\frac{1}{2}$185$m_{j}\in \mathbb{Z}_{\geq 0}(j=1, \cdots, d)_{f}$we have$\frac{B_{p1}\cdots B_{pd}}{p_{1}!\cdots p_{d}!}\cross$$\zeta_{d}(-m_{1}+\epsilon_{1}, \cdots, -m_{d}+\epsilon_{d})=(-1)^{m_{d}}m_{d}!$$\sum_{p_{1}+\cdots+p_{d}=d+M}$$p_{1},\cdots,p_{d}\geq 0$$-md^{-m_{d}(j)-d+j+pd(j)<2or}(j-1)-d+j+p_{d}(j)\geq 2(2\leq\forall j\leq d)$$\cross\prod_{j=2}^{d}\frac{[\epsilon_{d}(j)]_{-m(j)-d+j+p_{d}(j)-1}d}{[\epsilon_{d}(j-1)]_{-m(j-1)-d+j+p_{d}(j)-1}d}+\sum_{j=1}^{d}O(\epsilon_{j})$as$(\epsilon_{1}, \cdots, \epsilon_{d})arrow(0, \cdots, 0)$, where$M:=m_{1}+\cdots+m_{d},$$[a]_{n}:=\{\begin{array}{ll}a(n-1)!

(n\geq 1) ,(-1)^{n}(-n)!^{-1} (n<1) .\end{array}$In the theorem, $\epsilon_{j}(j=1, \cdots, d)$ should satisfy $|\epsilon_{k}/\epsilon_{d}(j)|\ll 1(j=1,$$k=$. Let us consider this condition. If, thentends to rapidly.
By (1.2), $s_{j}+\cdots+s_{d}=-M$ is a singular locus. Therefore, when
, the
point
approximates asymptotically to the singular locus. Hence,means geometrically that
does not approximate
asymptotically to the singular locus.
$\cdots,$$j,$ $\cdots,$ $d)$$|\epsilon_{k}/\epsilon_{d}(j)|arrow\infty$

$d,$

$0$

$\epsilon_{d}(j)$$|\epsilon_{k}/\epsilon_{d}(j)|arrow\infty$$(-m_{1}+\epsilon_{1}, \cdots , -m_{d}+\epsilon_{d})$$|\epsilon_{k}/\epsilon_{d}(j)|\ll 1$3$(-m_{1}+\epsilon_{1}, \cdots, -m_{d}+\epsilon_{d})$ExamplesBy the main theorem, we can compute various multiple zeta values at non-positive integers. Let us see some examples.
In the case $d=2$ , we have,$\zeta_{2}(\epsilon_{1}, \epsilon_{2})=\frac{1}{3}+\frac{1}{12}\cdot\frac{\epsilon_{2}}{\epsilon_{1}+\epsilon_{2}}+\sum_{j=1}^{2}O(\epsilon_{j})$,$\zeta_{2}(-1+\epsilon_{1}, \epsilon_{2})=\frac{1}{24}+\sum_{j=1}^{2}O(\epsilon_{j})$,$\zeta_{2}(\epsilon_{1}, -1+\epsilon_{2})=\frac{1}{12}+\sum_{j=1}^{2}O(\epsilon_{j})$$\zeta_{2}(-1+\epsilon_{1}, -1+\epsilon_{2})=\frac{1}{360}+\frac{1}{720}\cdot\frac{\epsilon_{2}}{\epsilon_{1}+\epsilon_{2}}+\sum_{j=1}^{2}O(\epsilon_{j})$.

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In the case $d=3$ , we have$\zeta_{3}(\epsilon_{1}, \epsilon_{2}, \epsilon_{3})=-\frac{1}{4}-\frac{1}{24}\cdot\frac{\epsilon_{3}}{\epsilon_{2}+\epsilon_{3}}-\frac{1}{24}\cdot\frac{\epsilon_{2}+2\epsilon_{3}}{\epsilon_{1}+\epsilon_{2}+\epsilon_{3}}+\sum_{j=1}^{3}O(\epsilon_{j})$$\zeta_{3}(-1+\epsilon_{1}, \epsilon_{2}, \epsilon_{3})=-\frac{17}{720}-\frac{1}{144}.

\frac{\epsilon_{3}}{\epsilon_{2}+\epsilon_{3}}+\frac{1}{720}.

$\zeta_{3}(\epsilon_{1}, -1+\epsilon_{2}, \epsilon_{3})=-\frac{19}{360}+\frac{1}{360}.

,,\epsilon_{1}+\epsilon_{2}+\epsilon_{3}-\epsilon_{2}+3\epsilon_{3}+\sum_{j=1}^{3}O(\epsilon_{j})$,\epsilon_{1}+\epsilon_{2}+\epsilon_{3}\epsilon_{2}+\sum_{j=1}^{3}O(\epsilon_{j})$$\zeta_{3}(\epsilon_{1}, \epsilon_{2}, -1+\epsilon_{3})=--\underline{3}\underline{1}.

4\epsilon_{2}+3\epsilon_{3}$$40 720 \epsilon_{1}+\epsilon_{2}+\epsilon_{3}+\sum_{j=1}^{3}O(\epsilon_{j})$.

Note that the example (1.5) comes from the first example of the above, taking
andIn the case $d=4$ , we have$\epsilon_{1}=\epsilon^{2}$$\cdot$$\epsilon_{2}=\epsilon_{3}=\epsilon.$$\epsilon_{4}$$\zeta_{4}(\epsilon_{1}, \epsilon_{2}, \epsilon_{3}, \epsilon_{4})=^{\underline{1}_{+}\underline{1}}.$$\epsilon_{3}+2\epsilon_{4}$$+\underline{1}.$$5 36 \epsilon_{3}+\epsilon_{4} 48 \epsilon_{2}+\epsilon_{3}+\epsilon_{4}$$+\underline{1}.

19\epsilon_{2}+33\epsilon_{3}+52\epsilon_{4}+\underline{1}.

\epsilon_{4}(\epsilon_{2}+\epsilon_{3}+\epsilon_{4})$$720 \epsilon_{1}+\epsilon_{2}+\epsilon_{3}+\epsilon_{4} 144 (\epsilon_{3}+\epsilon_{4})(\epsilon_{1}+\epsilon_{2}+\epsilon_{3}+\epsilon_{4})$$+ \sum_{j=1}^{4}O(\epsilon_{j})$4.

Proof of the Main Theorem

is the Riemann zeta function.
In this section, we prove the main theorem. If $d=1,$Hence, the main theorem is clear. So we prove the theorem in the case $d>1.$
has an.
First, we prove the meromorphic continuation of
integral representation as the following,$\zeta_{1}(s_{1})$$\zeta_{d}(s_{1}, \cdots, s_{d})$$\zeta_{d}(s_{1}, \cdots, s_{d})$$\Gamma(s_{1})\cdots\Gamma(s_{d})\zeta_{d}(s_{1}, \cdots, s_{d})$$= \int_{0}^{1}\ldots\int_{0}^{1}\int_{0}^{\infty}dj-1.$ (4.1)

187
Dividing the integral into two parts and integrating by parts, (4.1) can be written$\zeta_{d}(s_{1}, \cdots, s_{d})=\frac{1}{\Gamma(s_{1})\cdots\Gamma(s_{d})}\sum_{k=0p_{1+}}^{n1}\sum_{+p_{d}=k}\frac{B_{p_{1}}\cdots B_{p_{d}}1}{p_{1}!\cdots p_{d}!\mathcal{S}_{d}(1)-d+k}\cross$$\cross\prod_{j=2}^{d}B(s_{d}(j)-d+j+p_{d}(j)-1, s_{j-1})$$+ \frac{1}{\Gamma(s_{1})\cdots\Gamma(s_{d})}\int_{0}^{1}x^{s_{d}(1)-d+n1}F_{\varphi}(x, n_{2}, \cdots, n_{d})dx$$+ \frac{1}{\Gamma(s_{1})\cdots\Gamma(s_{d})}\int_{1}^{\infty}\frac{x^{s(1)-d}d}{e^{x}-1}F_{\psi}(x, n_{2}, \cdots, n_{d})dx$

where$n_{1},$$\cdots,$$n_{d}\in \mathbb{Z}_{\geq 0}.$$(4.2)$,(4.2)

can be continued meromorphically to$\{(s_{1}, \cdots, s_{d})\in \mathbb{C}^{d}|\Re(s_{j-1})>-n_{j}-1(j=2,\cdot d)\Re(s_{d}(j))>d-j-n_{j}(j=.1,.\cdot,\cdot\cdot, d), \}\cdot$

We use (4.2) with $s_{j}=-m_{j}+\epsilon_{j}(j=1, \cdots, d)$ and $n_{1}=\cdots=n_{d}=M+d$ . By
estimating the second term and the third terms of (4.2), we obtain.

$\frac{1}{\Gamma(s_{1})\cdots\Gamma(s_{d})}(\int_{0}^{1}x^{\mathcal{S}}d(1)-d+n1F_{\varphi}(x)dx+l^{\infty}\frac{x^{s(1)-d}d}{e^{x}-1}F_{\psi}(x)dx)=\sum_{j=1}^{d}O(\epsilon_{j})$Consider the first term of (4.2) by writing it as the following,$d$

ハイキみ 1$\sum=$$\sum$

ルノキ$+$$\sum$

,

$($

4.3

$)$

$k=0 k=0 k=M+d$and estimating the first term of (4.3), we obtain
ルプキ $d-1$

ヨ$\sum_{k=0} =\sum_{j=1}O(\epsilon_{j})$.

Next, we consider the second term of (4.3). First, we estimate the factors containing
gamma funcions and beta functions as the following,$\frac{1}{\Gamma(s_{1})\cdots\Gamma(s_{d})}\prod_{j=2}^{d}B(s_{d}(j)-d+j+p_{d}(j)-1, \mathcal{S}_{j-1})$$= \frac{1}{\Gamma(s_{d})}\prod_{j=2}^{d}\frac{\Gamma(s_{d}(j)-d+j+p_{d}(j)-1)}{\Gamma(s_{d}(j-1)-d+j+p_{d}(j)-1)}$$= \frac{1}{(\epsilon_{d})_{-m_{d}}\Gamma(\epsilon_{d}(1))}\prod_{j=2}^{d}\frac{(\epsilon_{d}(j))_{m(j)-d+j+p_{d}(j)-1}d}{(\epsilon_{d}(j-1))_{m(j-1)-d+j+p_{d}(j)-1}d}$

.

(4.4)

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Finally, to remove the function
obtain the main theorem.

$h$from (4.6), we restrict the summation. Then weReferences[1] S. Akiyama, S. Egami and Y. Tanigawa, Analytic continuation of multiple zetafunctions and their values at non-positive integers, Acta Arith, 98 (2001), 107-116.
[2] S. Akiyama, Y. Tanigawa, Multiple zeta values at non-positive integers, Ramanujan
5 (2001), 327-351.

$J.$[3] Y. Komori, An integml representation of multiple Hurwitz-Lerch zeta functions andgeneralized multiple bernoulli numbers, Quart. J. Math. (Oxford) (2009), 1-60.

[4] K. Matsumoto, On analytic continuation of various multiple zeta-functions, Number
Theory for the Millenium (Urbana, 2000), Vol. II, M. A. Bennett et. al. (eds.), A. K.
Peters, Natick, $MA$ , 2002, pp. 417-440.
[5] K. Matsumoto, The analytic continuation and the asymptotic behaviour of certain
multiple zeta-functions , J. Number Theory 101 (2003), 223-243.
$I$[6] Y. Sasaki, Multiple zeta values for coordinatewise limits at non-positive integers, Acta
Arith. 136 (2009), 299-317.
[7] J. Zhao, Analytic continuation of multiple zeta functions, Proc. Amer. Math. Soc. 128(2000), 1275-1283.

Graduate School of Mathematics
Nagoya University
Chikusa-ku, Nagoya 464-8602, JapanE–mail: m11022v@math.nagoya-u.ac.jp