\({\varOmega}_\pm\)-results of the error term in the mean square formula of the Riemann zeta-function in the critical strip (Q2717614)

For \(1/2< \sigma< 3/4\) let NEWLINE\[NEWLINE\int_0^T |\zeta (\sigma+it)|^2 dt= \zeta(2\sigma)T+ (2\pi)^{2\sigma-1} \frac{\zeta(2-2\sigma)} {(2-2\sigma)}+ E_\sigma(T).NEWLINE\]NEWLINE It has been shown by \textit{K. Matsumoto} and \textit{T. Meurman} [Acta Arith. 64, 357-382 (1993; Zbl 0788.11035)] that NEWLINE\[NEWLINEE_\sigma(T)= \Omega_+ (T^{3/4-\sigma} (\log T)^{\sigma-1/4}),NEWLINE\]NEWLINE and by \textit{A. Ivić} and \textit{K. Matsumoto} [Monatsh. Math. 121, 213-229 (1996; Zbl 0843.11039)] that NEWLINE\[NEWLINEE_\sigma(T)= \Omega_- (T^{3/4-\sigma} \exp(C(\log T)^{\sigma-1/4} (\log\log T)^{\sigma-5/4})),NEWLINE\]NEWLINE for some positive constant \(C\). NEWLINENEWLINENEWLINEThe present paper sharpens this latter result, proving NEWLINE\[NEWLINEE_\sigma(T)= \Omega_\pm (T^{3/4-\sigma} (\log T)^{\sigma-1/4}).NEWLINE\]NEWLINE It is interesting to note that the proof breaks down at \(\sigma= 1/2\). For \(\sigma= 3/4\) it is shown that NEWLINE\[NEWLINE\sup_{t\in [T,T+ L\sqrt{T}]} \pm E_{3/4}(t)\gg \sqrt{\log T},NEWLINE\]NEWLINE so that firstly \(E_{3/4}(T)= \Omega_\pm ((\log T)^{1/2})\), and secondly, \(E_{3/4} (T)\) changes sign in every interval of length \(\gg \sqrt{T}\).