# Moments of $S(T)$

is defined by

where the argument is obtained by continuous variation from where the argument is 0, to to , circumventing zeros of by small semicircular detours above the zeros. Selberg [ MR 8,567e] proved an asymptotic formula for

for positive integral values of and an appropriate . Goldston [ MR 89a:11086], assuming the Riemann Hypothesis, was able to give a second main term in the case that . Keating and Snaith's conjectures for moments of imply formula for the above moments of , including lower order terms all the way to a constant, i.e. they conjecture that

for some explicit constants .

It seems like further work should allow one to obtain the lower order terms in the moments of ; 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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