Sums of k-th powers and the Whittaker–Fourier coefficients of automorphic forms

Abstract

In this work, we obtain power-saving bounds for shifted convolution sums involving the Whittaker–Fourier coefficients of automorphic forms and $$r_{s, k}(n)$$ , the number of representations of a positive integer n as a sum of $$s\;k$$ -th positive integral powers, based on the recently proved Main Conjecture in Vinogradov’s Mean Value Method.