We prove an asymptotic formula for ∑_n≤N r(n)r(n+m) using the spectral theory of automorphic forms and we specially study the uniformity of the error them 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 inequalinties for Fourier coefficients of cusp forms to estimate some averages (over m) of the error term.
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