High power sums of Fourier coefficients of holomorphic cusp forms and their applications

Abstract

Let $ \lambda_f(n) $ be the $ n $th normalized Fourier coefficient of a holomorphic cusp form $ f $ for the full modular group. In this paper, we established asymptotic formulae for high power sums of Fourier coefficients of cusp forms and further improved previous results. Moreover, as an application, we studied the signs of the sequences $ \{\lambda_f(n)\} $ and $ \{\lambda_f(n)\lambda_g(n)\} $ in short intervals, and presented some quantitative results for the number of sign changes for $ n\leq x $.

Keywords

• cusp form
• $ l $-function
• fourier coefficient
• sign change