Correlated sums of r(n)

Abstract

We prove an asymptotic formula for \displaystyle ∑_{n≤ N}r(n)r(n+m) using the spectral theory of automorphic forms and we specially study the uniformity of the error term in the asymptotic approximation when m varies. The best results are obtained under a natural conjecture about the size of a certain spectral mean of the Maass forms.
We also employ large sieve type inequalities for Fourier coefficients of cusp forms to estimate some averages (over m) of the error term.