Extreme values for $S_n(\sigma,t)$ near the critical line

Abstract

Let $S(\sigma,t)=\frac{1}{\pi}\arg\zeta(\sigma+it)$ be the argument of the Riemann zeta function at the point $\sigma+it$ of the critical strip. For $n\geq 1$ and $t>0$ we define $$ S_{n}(\sigma,t) = \int_0^t S_{n-1}(\sigma,\tau)\,d\tau\, + \delta_{n,\sigma\,}, $$ where $\delta_{n,\sigma}$ is a specific constant depending on $\sigma$ and $n$. Let $0\leq \beta<1$ be a fixed real number. Assuming the Riemann hypothesis, we show lower bounds for the maximum of the function $S_n(\sigma,t)$ on the interval $T^\beta\leq t \leq T$ and near to the critical line, when $n\equiv 1\mod 4$. Similar estimates are obtained for $|S_n(\sigma,t)|$ when $n\not\equiv 1\mod 4$. This extends a recently results of Bondarenko and Seip for a region near the critical line. In particular we obtain some omega results for these functions on the critical line.