On the means of the argument of the Riemann zeta-function on the critical line (Q1187287)

Let \(S(t)=\arg\zeta(1/2+it)\) where the argument of the Riemann zeta- function \(\zeta(s)\) is defined appropriately. The main result proved in this paper is the following Theorem: Let \(H\) be a function of \(T\) such that \(T^ \alpha\leq H(T)\leq T\) where \(1/2<\alpha\leq 1\). Then, for any positive integer \(k\) \[ \int^{T+H}_ T| S(t)|^ k dt\sim{2^ k\over\sqrt\pi} \Gamma\left({k+1\over 2}\right)\left({1\over 2\pi}\right)^ kH(\log\log T)^{k/2},\quad T\to\infty. \] This generalizes the corresponding result of \textit{A. Ghosh} [J. Number Theory 17, 93-102 (1983; Zbl 0511.10030)] who proved the above asymptotic formula for \(k=1\) and an even \(k\).