Quadratic forms connected with Fourier coefficients of holomorphic and Maass cusp forms

TL;DR

This paper proves a prime number theorem related to Fourier coefficients of cusp forms using the circle method, improving estimates by leveraging recent advances on Siegel zeros for $GL(2)$ L-functions.

Contribution

It introduces a prime number theorem for Fourier coefficients of cusp forms, utilizing the circle method and recent non-existence results of Siegel zeros to enhance previous bounds.

Findings

Established a prime number theorem for Fourier coefficients of cusp forms.

Improved estimates based on recent results about Siegel zeros.

Applied the circle method to analyze Fourier coefficients in this context.

Abstract

In this work we prove a prime number type theorem involving the normalised Fourier coefficients of holomorphic and Maass cusp forms, using the classical circle method. A key point is in a recent paper of Fouvry and Ganguly, based on Hoffstein-Ramakrishnan's result about the non-existence of the Siegel zeros for $GL(2)$ $L$-functions, which allows us to improve preceding estimates.