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

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 $a_p{b}_p\ne 0$; simultaneous non-vanishing in the short intervals and in arithmetic progressions; simultaneous sign changes in short intervals.

11 pages, To appear Int. J. Number Theory