The second moment of $S_n(t)$ on the Riemann hypothesis

Abstract

Let $S(t)=\frac{1}{\pi }\arg \zeta (1/2+it)$ be the argument of the Riemann zeta-function at the point $\frac{1}{2}+it$. For $n\ge 1$ and $t>0$ define its antiderivatives as \begin{equation*} S_n(t) = \int_0^t S_{n-1}(τ) \hspace{0.08cm} \rm dτ+ δ_n, \end{equation*} where ${\delta }_n$ is a specific constant depending on $n$ and $S_0(t):=S(t)$. In 1925, J. E. Littlewood proved, under the Riemann Hypothesis, that $${\int }_0^T|S_n(t){|}^2\mathrm{dt}=\mathrm{O}(\mathrm{T}),$$ for $n\ge 1$. In 1946, Selberg unconditionally established the explicit asymptotic formulas for the second moments of $S(t)$ and $S_1(t)$. This was extended by Fujii for $S_n(t)$, when $n\ge 2$. Assuming the Riemann Hypothesis, we give the explicit asymptotic formula for the second moment of $S_n(t)$ up to the second-order term, for $n\ge 1$. Our result conditionally refines Selberg's and Fujii's formulas and extends previous work by Goldston in 1987, where the case $n=0$ was considered.

arXiv commentTo appear in Int. J. Number Theory