Generations of correlation averages

TL;DR

This paper introduces an elementary approach to bound the Selberg integral of the divisor function d_3, linking it to correlation averages, and extends results to higher divisor functions with implications for the moments of the Riemann zeta function.

Contribution

It provides a new elementary Dispersion Method-based framework to estimate Selberg integrals and generalizes bounds to divisor functions d_k for k≥3, improving prior results.

Findings

Established non-trivial bounds for the Selberg integral of d_3.

Linked weighted Selberg integrals to correlation averages of arithmetic functions.

Extended results to divisor functions d_k for any k≥3.

Abstract

The present paper is a dissertation on the possible consequences of a conjectural bound for the so-called \thinspace modified Selberg integral of the divisor function $d_3$, i.e. a discrete version of the classical Selberg integral, where $d_3(n)=\sum_{abc=n}1$ is attached to the Cesaro weight $1-|n-x|/H$ in the short interval $|n-x|\le H$. Mainly, an immediate consequence is a non-trivial bound for the Selberg integral of $d_3$, improving recent results of Ivi\'c based on the standard approach through the moments of the Riemann zeta function on the critical line. We proceed instead with elementary arguments, by first applying the "elementary Dispersion Method" in order to establish a link between "weighted Selberg integrals" \thinspace of any arithmetic function $f$ and averages of correlations of $f$ in short intervals. Moreover, we provide a conditional generalization of our results to the analogous problem on the divisor function $d_k$ for any $k\ge 3$. Further, some remarkable consequences on the $2k-$th moments of the Riemann zeta function are discussed. Finally, we also discuss the essential properties that a general function $f$ should satisfy so that the estimation of its Selberg integrals could be approachable by our method.