## 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 $I_T(s_0, s)=\int\limits_2^T\left|\zeta_2(s_0, s)\right|^2 dt,$
$\zeta_2^{}(s_0, s)=\sum_{k=2}^\infty\left|\sum_{m=1}^{k-1}{1\over m^{s_0}}\right|{1\over k^s}$ and $R_T(s_0, s)=I_T(s_0, s)-\zeta_2^{}(s_0, s).$ 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.

### MSC:

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

Zbl 0036.18603
Full Text:

### References:

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