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

TL;DR

This paper establishes an upper bound for a modified Selberg integral of the three-divisor function, providing insights into its behavior in short intervals and implications for the weak sixth moment of the Riemann zeta function.

Contribution

It introduces a new bound for the modified Selberg integral of d_3(n), advancing understanding of divisor functions and their moments.

Findings

Proves $ ext{modSel}_3(N,h) \\ll Nh^2L^2$ bound.

Provides evidence towards the weak sixth moment of the Riemann zeta function.

Suggests methods for future proofs of related moments.

Abstract

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.(This was OLD abstract)