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

Abstract

Let $S(σ,t)=\frac{1}π\argζ(σ+it)$ be the argument of the Riemann zeta-function at the point $σ+it$ in the critical strip. For $n\geq 1$ and $t>0$, we define \begin{equation*} S_{n}(σ,t) = \int_0^t S_{n-1}(σ,τ) \,dτ+ δ_{n,σ\,}, \end{equation*} where $δ_{n,σ}$ is a specific constant depending on $σ$ and $n$. Let $0\leq β<1$ be a fixed real number. Assuming the Riemann hypothesis, we establish lower bounds for the maximum of $S_n(σ,t+h)-S_n(σ,t)$ near the critical line, on the interval $T^β\leq t \leq T$ 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.