Average behavior of Fourier coefficients of cusp forms

Abstract

Let a 0 ( n ) a_0(n) and b 0 ( n ) b_0(n) be the normalized Fourier coefficients of the two holomorphic Hecke eigenforms f ( z ) ∈ S 2 k ( Γ ) f(z)\in S_{2k}(\Gamma ) and φ ( z ) ∈ S 2 l ( Γ ) \varphi (z)\in S_{2l}(\Gamma ) respectively. In 1999, Fomenko studied the following average sums of a 0 ( n ) a_0(n) and b 0 ( n ) b_0(n) : \[ ∑ n ≤ x a 0 ( n ) 3 , ∑ n ≤ x a 0 ( n ) 2 b 0 ( n ) , ∑ n ≤ x a 0 ( n ) 2 b 0 ( n ) 2 , ∑ n ≤ x a 0 ( n ) 4 . \sum _{n \leq x}a_0(n)^3, \quad \sum _{n \leq x}a_0(n)^2b_0(n), \quad \sum _{n \leq x}a_0(n)^2b_0(n)^2, \quad \sum _{n \leq x}a_0(n)^4. \] In this paper, we are able to improve on Fomenko’s results.