Sign changes and nonvanishing of Fourier coefficients of holomorphic cusp forms

Abstract

We focus on the problems of sign changes and nonvanishing of Fourier coefficients of holomorphic cusp forms. Let λf(n), λg(n) and λh(n) be the n-th Fourier coefficients of primitive holomorphic cusp forms f, g and h for SL(2,ℤ), respectively. Firstly, we study the behavior of the signs of the sequences {λf(nj)} with j≥3, {λf(ni)λg(nj)} with i≥1,j≥2, and {λf(n)λg(n)λh(n)} in short intervals, and present some quantitative results for the number of sign changes for n≤x. Here we improve and generalize previous results. Moreover, we investigate simultaneous nonvanishing of {λf(nj)λg(nj)λh(nj)} with j≥1 in short intervals.