Godement-Jacquet $L$-function, some conjectures and some consequences

TL;DR

This paper explores the properties of the Godement-Jacquet $L$-function, focusing on mean square estimates of its logarithmic derivative under the Riemann hypothesis and Rudnick--Sarnak conjecture.

Contribution

It provides new insights into the behavior of the Godement-Jacquet $L$-function's logarithmic derivative assuming key conjectures.

Findings

Mean square estimates derived under RH and Rudnick--Sarnak conjecture

Conditional results on the distribution of zeros of $L_f(s)$

Connections between conjectures and $L$-function properties

Abstract

In this paper, we investigate the mean square estimate for the logarithmic derivative of the Godement--Jacquet $L$-function $L_f(s)$ assuming the Riemann hypothesis for $L_f(s)$ and Rudnick--Sarnak conjecture.