## On the mean square of the Riemann zeta-function.(English)Zbl 0624.10032

Let as usual $$E(T)=\int^{T}_{0}| \zeta (+it)|^ 2 dt-T(\log (T/2\pi)+2\gamma -1)$$, where $$\gamma$$ is Euler’s constant. A sophisticated averaging technique is used to derive $(1)\quad E(T)=\mathbf{\Sigma}^*_ 1(T)+\mathbf{\Sigma}^*_ 2(T)+\pi +O(T^{- 1/4}\quad \log T),$ where the sums $$\Sigma$$ $${}^*_ j(T)$$ (whose length is of order T) correspond to the sums $$\Sigma$$ $${}_ 1(T)$$ and $$\Sigma$$ $${}_ 2(T)$$ in F. V. Atkinson’s formula [Acta Math. 81, 353-376 (1949; Zbl 0036.186)] for E(T). Squaring (1) and integrating termwise the author obtains $(2)\quad \int^{T}_{2}E^ 2(t) dt=cT^{3/2}+O(T \log^ 5 T)\quad (c=(2/3)(2\pi)^{-}\zeta^ 4(3/2)/\zeta (3)),$ which improves the earlier result of D. R. Heath-Brown [Mathematika 25, 177-184 (1979 Zbl 0387.10023)], who had the weaker error term $$O(T^{5/4} \log^ 2 T)$$. Y. Motohashi [Proc. Japan Acad., Ser. A 62, 311-313 (1986; Zbl 0613.10032)] proved independently (2) by a method different from the author’s.
The present paper ends with a new, simple proof of the analogue of (2) when E(t) is replaced by $$\Delta$$ (t) (the error term in the classical divisor problem).
Reviewer: A.Ivić

### MSC:

 11M06 $$\zeta (s)$$ and $$L(s, \chi)$$ 11N37 Asymptotic results on arithmetic functions

### Citations:

Zbl 0396.10025; Zbl 0036.186; Zbl 0387.10023; Zbl 0613.10032
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