On large values of the function \(S(t)\) on short intervals (Q650449)

Here the author proves a theorem on upper and lower bounds for the argument \(S(t)\) of the Riemann zeta function on short intervals of the critical line. The theorem proves is a similar statement of the following theorem of \textit{M. A. Korolev} [Izv. Math. 69, No. 1, 113--122 (2005); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 69, No. 1, 115--124 (2005; Zbl 1167.11313)]: Theorem 1. Suppose that \(T > T_0 > 0\), \((\ln T)(\ln\ln T)-3/2 < H < T\). If the Riemann hypothesis is valid, then the following inequalities hold: \[ \sup_{T-H\leq t\leq T+2H} (\pm S(t))\geq \frac 1{90\pi}\sqrt{\frac{\ln H}{\ln\ln H}}. \] The present result with still smaller value of \(H\) is as follows: Theorem 2. If the Riemann hypothesis is valid, then there exists an absolute positive constant \(T_1\) such that, for all real numbers \(T > T_1\) and \(\sqrt{\ln\ln H} \leq H \leq (\ln T)(\ln \ln T)-3/2\), the following inequalities hold: \[ \sup_{T-H\leq t\leq T+2H} (\pm S(t))\geq \frac 1{900}\frac{\sqrt{\ln H}}{\ln\ln H}. \]