Higher moments of fourier coefficients of cusp forms

Abstract

Let SA.(F) be the space of holomorphic cusp forms of even integral weight k for the full modular group SL(2,Z). Let A/(), Xg(n), A/,(n) be the n-th normalized Fourier coefficients of three distinct holomorphic primitive cusp forms f(z) € Skl(r),g(z) e S∗,(r), and h(z) e Sfc3( T), respectively. In this paper we study the cancellations of sums related to arithmetic functions, such as A f(n)4Ag(n)2, As(n)6, As(n)2Ah(n)4, and Ag(n3)2 twisted by the arithmetic function A/(n).