An Additive Problem in the Fourier Coefficients of Cusp Forms

s=1/2 where E(z, s) is the Eisenstein series for SL2(Z). In general one tries to deduce good estimates for these sums assuming the parameters a, b, h are of considerable size. The additive divisor problem has an extensive history and we refer the reader to [1] for a short introduction. Let us just mention that in the special case a = b = 1 one can derive very sharp results by employing the spectral theory of automorphic forms for the full modular group. This approach is hard to generalize for larger values of a, b as one faces difficulties with small Laplacian eigenvalues for the congruence subgroup Γ0(ab). The idea of Duke, Friedlander and Iwaniec [1] is to combine the more elementary δ-method (a variant of the classical circle method) with a Voronoi-type summation formula for the divisor function and then apply Weil’s estimate for the individual Kloosterman sums