On the Selberg integral of the $k-$divisor function and the $2k-$th moment of the Riemann zeta function

TL;DR

This paper explores the connection between the $2k$-th moments of the Riemann zeta function and the Selberg integral of the $k$-divisor function, using elementary arithmetic average methods to deepen understanding of their relationship.

Contribution

It introduces an elementary approach based on arithmetic averages to analyze the link between zeta moments and divisor functions, extending previous bounds and insights.

Findings

Established new bounds on the Selberg integral of $d_k$

Provided a reverse link analysis to previous zeta moment bounds

Enhanced understanding of the relationship between zeta moments and divisor functions

Abstract

We deeply appreciate the papers of Ivi\'c on the links between the $2k-$th moments of the Riemann zeta function and, say, $d_k$, the $k-$divisor function. More specifically, both the one bounding the $2k-$th moment with a simple average of correlations of the $d_k$ (Palanga 1996 Conference Proceedings) and the more recent (arXiv:0708.1601v2 to appear on JTNB), which bounds the Selberg integral of $d_k$ applying the $2k-$th moment of the zeta. Building on the former paper, we apply an elementary approach (based on arithmetic averages) in order to get information on the reverse link to the second work.