We prove an asymptotic formula for using the spectral theory of automorphic forms and we specially study the uniformity of the error term in the asymptotic approximation when 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 ) of the error term.
Fernando CHAMIZO. "Correlated sums of ." J. Math. Soc. Japan 51 (1) 237 - 252, January, 1999. https://doi.org/10.2969/jmsj/05110237