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 $n\ge 1$ 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 $0\le \beta <1$ 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 $T^{\beta }\le t\le T$ and near to the critical line, when $n\equiv 1\mathrm{mod}4$. Similar estimates are obtained for $|S_n(\sigma ,t)|$ when $n\not\equiv 1\mathrm{mod}4$. 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.