Mean value theorems for the double zeta-function. (English) Zbl 1355.11088

Matsumoto, Kohji; Tsumura, Hirofumi

J. Math. Soc. Japan 67, No. 1, 383-406 (2015).

Let \(s_1\) and \(s_2\) be complex variables. The Euler double zeta-function \(\zeta_2(s_1, s_2)\) is defined, for \(\operatorname{Re} (s_2)>1\) and \(\operatorname{Re}(s_1+s_2)>2\), by \[ \zeta_2(s_1, s_2)=\sum_{m=1}^\infty{1\over m^{s_1}}\sum_{n=1}^\infty{1\over (m+n)^{s_1}}=\sum_{k=2}^\infty\left(\sum_{m=1}^{k-1}{1\over m^{s_1}}\right){1\over k^{s_2}}, \] and is meromorphically continued to \(\mathbb{C}^2\) with singularities at \(s=1\) and \(s_1+s_2=2, 1, 0, -2, -4, \dots\). The function \(\zeta_2(s_1, s_2)\) was widely studied by many authors, including F. V. Atkinson [Acta Math. 81, 353–376 (1949; Zbl 0036.18603)], the authors of the paper under review and their Japanese colleagues. They discovered the analytic continuation for \(\zeta_2(s_1, s_2)\), functional equations and various estimates. The present paper is devoted to the mean square formulae for \(\zeta_2(s_1, s_2)\) in various regions. Let \[ I_T(s_0, s)=\int\limits_2^T\left|\zeta_2(s_0, s)\right|^2 dt, \]
\[ \zeta_2^{[2]}(s_0, s)=\sum_{k=2}^\infty\left|\sum_{m=1}^{k-1}{1\over m^{s_0}}\right|{1\over k^s} \] and \[ R_T(s_0, s)=I_T(s_0, s)-\zeta_2^{[2]}(s_0, s). \] Then it is proved that if \(s_0=\sigma_0+it\), \(\sigma_0>1\), and \(s=\sigma+it\), \(\sigma>1\), \(t\geq 2\), then \(R_T(s_0, s)=O(1)\); if \(\sigma_0>1\), \({1\over 2}<\sigma<1\), \(t\geq 2\), and \(\sigma_0+\sigma>2\), then \(R_T(s_0, s)=O\left(T^{2-2\sigma}\log T\right)+O\left(T^{1/2}\right)\).
The main result of the paper is a series of estimates. Suppose that \({3\over 2}<\sigma_0+\sigma\leq 2\) and that when \(t\) moves from 2 to \(T\), the point \((s_0, s)\) does not encounter the hyperplane \(s_0+s=2\). Then \[ R_T(s_0, s) = \begin{cases} O\left(T^{4-2\sigma_0-2\sigma}\log T\right)+O\left(T^{1/2}\right), &\text{ if }{1\over 2}<\sigma_0<1,\quad {1\over 2}<\sigma<1, \\ O\left(T^{2-2\sigma}\log^2 T\right)+O\left(T^{1/2}\right), &\text{ if }{1\over 2}<\sigma_0<1,\quad \sigma=1, \\ O\left(T^{2-2\sigma}\log^3 T\right)+O\left(T^{1/2}\right), & \text{ if }\sigma_0=1,\quad {1\over 2}<\sigma<1, \\ O\left(T^{1/2}\right), &\text{ if }\sigma_0=1,\quad \sigma=1, \\ O\left(T^{2-2\sigma}\log T\right)+O\left(T^{1/2}\right), &\text{ if }1<\sigma_0<{3\over 2},\quad {1\over 2}<\sigma<1. \end{cases} \]
Also, the relation of the obtained estimates to an analogue of the Lindelöf hypothesis is shortly discussed.

Reviewer: Renata Macaitiene (Vilnius)

MSC:

11M32	Multiple Dirichlet series and zeta functions and multizeta values
11M06	\(\zeta (s)\) and \(L(s, \chi)\)

Keywords:

double zeta-functions; mean values; Lindelöf hypothesis; Euler’s constant

Citations:

Zbl 0036.18603

References:

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