On the Rankin-Selberg $L$-function related to the Godement-Jacquet $L$-function II

TL;DR

This paper derives an improved asymptotic formula for the Riesz means of coefficients of the Rankin-Selberg $L$-function related to the Godement-Jacquet $L$-function, enhancing understanding of their partial sums.

Contribution

It establishes an asymptotic formula with an improved range and error term for the Riesz means of these $L$-function coefficients, advancing analytic number theory techniques.

Findings

Asymptotic relation for partial sums of coefficients

Improved error term in Riesz mean asymptotics

Enhanced range for the asymptotic formula

Abstract

In this paper, we consider the $k$-th Riesz mean for the coefficients of the Rankin-Selberg $L$-function $L_{f \times f}(s)$ related to the Godement-Jacquet $L$-function with respect to $SL(n,\mathbb{Z})$. We establish an asymptotic formula for the $k$-th Riesz mean with an improved range and a better error term. As a result, we get an asymptotic relation for the partial sum of the coefficients of $L_{f \times f}(s)$.