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

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.


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


Zbl 0036.18603
Full Text: DOI arXiv Euclid


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