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

TL;DR

This paper investigates the extreme values of iterated argument functions related to the Riemann zeta function near the critical line, providing lower bounds and omega results under the Riemann hypothesis.

Contribution

It extends recent results by Bondarenko and Seip to a broader region near the critical line, establishing new lower bounds and omega results for iterated argument functions.

Findings

Lower bounds for maximum of $S_n(\sigma,t)$ near the critical line.

Omega results for $S_n(\sigma,t)$ on the critical line.

Extension of previous work to a larger region 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.