On the modified Selberg integral of the three-divisor function $d_3$

Details On the modified Selberg integral of the three-divisor function $d_3$ On the modified Selberg integral of the three-divisor function $d_3$

2011-06-25

arxiv.org/abs/1106.5696v3

Mathematics

Number Theory

Still, the Theorem holds; but square-root cancellation only in the Proposition. Problem's the variable range (not covered by P

Scientific paper

We prove a non-trivial result for the,say,modified Selberg integral $\modSel_3(N,h)$, of the divisor function $d_3(n):= \sum_{a}\sum_{b}\sum_{c, abc=n}1$; this integral is a slight modification of the corresponding Selberg integral, that gives the expected value of the function in short intervals. We get, in fact, $\modSel_3(N,h)\ll Nh^2L^2$, where $L:=\log N$; furthermore, as a byproduct, we obtain indications on the way in which it may be proved the weak sixth moment of the Riemann zeta function.

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