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

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.