Jutila's circle method and [formula omitted] shifted convolution sums

Abstract

Let λf(n) be the normalized Fourier coefficients of a Hecke-Maass or Hecke holomorphic cusp form f for congruence group Γ0(N) with level N and nebentypus χN. Let Q(x) be a positive definite integral quadratic form, and r(n,Q) denote the number of representations of n by the quadratic form Q. In this paper, we apply Jutila's circle method to derive a sharp bound for the shifted convolution sum of Fourier coefficients λf(n) and r(n,Q). We generalize and improve previous results without the Ramanujan conjecture.