The asymptotic behavior of the second power moment of the Riemann zeta-function in the critical strip
For \(1/2< \sigma< 1\) let, as usual, \[ E_\sigma (T) = \int_0^T \bigl|\zeta (\sigma+ it) \bigr|^2 dt- \zeta (2\sigma) T- {(2\pi)^{2 \sigma-1} \over 2-2 \sigma} \zeta(2-2 \sigma) T^{2-2 \sigma} \] denote the error term in the mean square formula for \(\zeta (\sigma +it)\). For this important function one conjectures that \(E_\sigma (T)\ll T^{\max (3/4-\sigma, 0) + \varepsilon}\) for \(1/2 < \sigma < 1\) fixed. Like in the work of \textit{A. Laurinčikas} [see the preceding review Zbl 0861.11050], the aim of this paper is to obtain uniform bounds, which are especially interesting when \(\sigma\) is close to \(1/2\). The author uses first the method of \textit{D. R. Heath-Brown} [Proc. London Math. Soc., III. Ser. 38, 385-422 (1979; Zbl 0403.10018)] to derive an expression for \(E_\sigma (T)\) (in weighted form). Then the method of \textit{D. R. Heath-Brown} and \textit{M. N. Huxley} [ibid. 61, 227-250 (1990; Zbl 0675.10027)] for the estimation of exponential sums with a difference is used, together with the theory of (one-dimensional) exponent pairs. The results are two theorems, of which only the first one will be stated here: If \(\sigma = 1/2+ \delta\), \(0< \delta < 1/2\), then uniformly for \(1/2< \sigma \leq 69/130\), \[ E_\sigma (T) \ll T^{{7\over 22} - {22 \over 165} \delta + \varepsilon}, \tag{1} \] and the factor ``\(T^\varepsilon\)'' can be replaced by a suitable log-power. \{Reviewer's remark: By using the more recent work of \textit{M. N. Huxley} [Bull. Lond. Math. Soc. 27, 325-327 (1994; Zbl 0853.11069)], which improves the exponent 7/22 of Heath-Brown and Huxley (op. cit.) to 72/227, one can replace the exponent in (1) by \(72/227- c \delta + \varepsilon\) with some explicit \(c>0\) in a range of \(\sigma\) close to \(1/2\}\).
scientific article; zbMATH DE number 939531