Moments of $S(T)$

$S(T)$ is defined by

\begin{displaymath}S(T)=\frac 1{\pi} \arg \zeta(1/2+iT)\end{displaymath}

where the argument is obtained by continuous variation from $s=2$ where the argument is 0, to $s=2+iT$ to $s=1/2+iT$, circumventing zeros of $\zeta(s)$ by small semicircular detours above the zeros. Selberg [ MR 8,567e] proved an asymptotic formula for

\begin{displaymath}\frac{1}{T}\int_0^T S(t)^{2k}~dt \sim c_k (\log \log T)^k\end{displaymath}

for positive integral values of $k$ and an appropriate $c_k$. Goldston [ MR 89a:11086], assuming the Riemann Hypothesis, was able to give a second main term in the case that $k=1$. Keating and Snaith's conjectures for moments of $\vert\zeta(1/2+it)\vert$ imply formula for the above moments of $S(T)$, including lower order terms all the way to a constant, i.e. they conjecture that

\begin{displaymath}\frac{1}{T}\int_0^T S(t)^{2k}~dt=\sum_{n=0}^k c_n (\log \log T)^n +o(1)\end{displaymath}

for some explicit constants $c_n=c_n(k)$.

It seems like further work should allow one to obtain the lower order terms in the moments of $S(T)$; it's possible that the assumption of the Riemann Hypothesis will allow for the evaluation of some of the lower order terms, and the assumption of GUE will allow for the rest.

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