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 $n\ge 1$ 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 ${\delta }_{n,\sigma }$ is a specific constant depending on $\sigma$ and $n$. Let $0\le \beta <1$ be a fixed real number. Assuming the Riemann hypothesis, we establish lower bounds for the maximum of $S_n(\sigma ,t+h)-S_n(\sigma ,t)$ near the critical line, on the interval $T^{\beta }\le t\le T$ 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.

To appear in Journal of Number Theory