Correlated sums of r(n)

Abstract

We prove an asymptotic formula for ${{\displaystyle \sum }}_{n\le N}r\left(n\right)r(n+m)$ 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.