A uniform estimate of the error term in the mean square of the Riemann zeta-function
For \(1/2 < \sigma < 1\) let \[ E_\sigma (T) = \int_0^T \bigl|\zeta (\sigma+ it) \bigr|^2 dt-\zeta (2\sigma) T- {(2\pi)^{2 \sigma-1} \over 2-2 \sigma} \zeta (2-2 \sigma) T^{2-2 \sigma} \] denote the error term in the mean square formula for \(\zeta (\sigma + it)\). This important function was introduced by \textit{K. Matsumoto} [Jap. J. Math., New Ser. 15, 1-13 (1989; Zbl 0684.10035)], and there exist several results concerning the order of \(E_\sigma (T)\) for various ranges of \(\sigma\) [e.g. \textit{K. Matsumoto} and the reviewer, Monatsh. Math. 121, 213-229 (1996; Zbl 0843.11039)]. The aim of the present paper [the results should be compared with those of \textit{A. Kačėnas}, see Zbl 0861.11051 below] is to obtain a uniform bound for \(E_\sigma (T)\), especially when \(\sigma\) is close to \(1/2\). If \(\delta = 2 \sigma-1, \beta_T = \min (1/ \delta, \log T)\), then for \(1/2 < \sigma<1\) one has \[ E_\sigma (T) \ll T^{1/(3+2 \delta)} \beta_T^{(4+4 \delta)/(3+2 \delta)} (\log T)^{(2+4 \delta)/(3+2\delta)}, \] which yields as a corollary the bound \[ E_\sigma (T) \ll T^{1/(3+2 \delta)} \log^2T \qquad (\delta \leq 1/ \log T). \] The method of proof is based on an explicit formula for \(E_\sigma (T)\), which is the analogue of \textit{F. V. Atkinson}'s classical formula for \(E(T)\) \((=\lim_{\sigma \to 1/2 + 0} E_\sigma (T))\) [Acta Math., Uppsala 81, 353-376 (1949; Zbl 0036.18603)]. This formula, which contains an error term of order \(\ll \beta_T \log T\) uniformly for \(1/2 < \sigma < 2/3\), is appropriately transformed by exponential averaging to yield (1). \{Reviewer's remark: Presumably slight improvements of (1) are possible by the use of the recent methods of M. N. Huxley and the theory of exponent pairs for the estimation of exponential sums\}.
scientific article; zbMATH DE number 939544