Oscillations of Fourier coefficients of product of L-functions at integers in a sparse set

Abstract

Let $f$ be a normalized Hecke eigenform of weight $k$ for the full modular group $S{L}_2(\mathbb{Z})$. In this paper, we obtain the asymptotic of higher moments of general divisor functions associated to the Fourier coefficients of Rankin–Selberg $L$-functions $R(s,f\times f)$, supported at the integers represented by primitive integral positive-definite binary quadratic forms (reduced forms) of a fixed discriminant $D<0.$ We improve previous results in the case when the reduced form is given by ${𝒬}(x_1,x_2)=x_1^2+x_2^2.$