Large values of the argument of the Riemann zeta-function and its iterates

TL;DR

This paper investigates the large values of the argument of the Riemann zeta-function and its iterates, establishing lower bounds and omega results under the Riemann hypothesis, thereby extending previous work and improving known bounds.

Contribution

It provides new lower bounds and omega results for the iterated argument functions of the Riemann zeta-function assuming the Riemann hypothesis, generalizing prior results.

Findings

Established lower bounds for maximum of iterated argument differences.

Derived omega results improving Selberg's bounds.

Extended previous results on the argument of the zeta-function.

Abstract

Let $S(\sigma,t)=\frac{1}{\pi}\arg\zeta(\sigma+it)$ be the argument of the Riemann zeta-function at the point $\sigma+it$ in the critical strip. For $n\geq 1$ and $t>0$, we define \begin{equation*} S_{n}(\sigma,t) = \int_0^t S_{n-1}(\sigma,\tau) \,d\tau + \delta_{n,\sigma\,}, \end{equation*} 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 establish lower bounds for the maximum of $S_n(\sigma,t+h)-S_n(\sigma,t)$ near the critical line, on the interval $T^\beta\leq t \leq T$ and in a small range of $h$. This improves some results of the first author and generalizes a result of the authors on $S(t)$. We also give new omega results for $S_n(t)$, improving a result by Selberg.