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


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