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

TL;DR

This paper introduces a new upper bound for the Selberg integral of the three-divisor function d_3(n), utilizing a novel conjecture, Laporta's method, and a modified Gallagher Lemma to advance understanding in analytic number theory.

Contribution

It presents a new non-trivial upper bound for the Selberg integral of d_3(n) using innovative conjectural and analytical techniques.

Findings

Established a new upper bound for the Selberg integral of d_3(n)

Applied a recent conjecture and Laporta's method in the analysis

Developed a modified Gallagher Lemma for exponential sums

Abstract

We give a new non-trivial upper bound for the Selberg integral of the three-divisor function $d_3(n)$. Our method applies our recent conjecture together with Laporta, for the modified Selberg integral of $d_3(n)$, and a kind of modified Gallagher Lemma for the exponential sums.