Simultaneous non-vanishing and sign changes of Fourier coefficients of modular forms

TL;DR

This paper investigates the behavior of Fourier coefficients of two modular forms, focusing on their simultaneous non-vanishing and sign changes across primes, short intervals, and arithmetic progressions.

Contribution

It provides new results on the existence of infinitely many primes with non-zero products and sign changes of Fourier coefficients of two modular forms.

Findings

Existence of infinitely many primes where both coefficients are non-zero.

Results on simultaneous non-vanishing in short intervals and arithmetic progressions.

Evidence of simultaneous sign changes in short intervals.

Abstract

In this article, we give some results on simultaneous non-vanishing and simultaneous sign-changes for the Fourier coefficients of two modular forms. More precisely, given two modular forms $f$ and $g$ with Fourier coefficients $a_n$ and $b_n$ respectively, we consider the following questions: existence of infinitely many primes $p$ such that $a_p b_p\neq 0$; simultaneous non-vanishing in the short intervals and in arithmetic progressions; simultaneous sign changes in short intervals.