Shifted convolution sums of divisor functions with Fourier coefficients

Abstract

Let $f\left(z\right)$ be a holomorphic cusp form of weight κ for the full modular group $S{L}_2\left(Z\right)$. Denote its n-th normalized Fourier coefficient by ${\lambda }_f\left(n\right)$. Let ${\tau }_k\left(n\right)$ denote that k-th divisor function with $k\ge 4$. In this paper, we consider the shifted convolution sum$\sum _{n\le X}{\tau }_k\left(n\right){\lambda }_f(n+h).$ We succeed in obtaining a non-trivial upper bound, which is uniform in the shift parameter h.