Triple Correlation Sums of Coefficients of Cusp Forms

Abstract

We produce nontrivial asymptotic estimates for shifted sums of the form $\sum a(h)b(m)c(2m-h)$, in which $a(n),b(n),c(n)$ are un-normalized Fourier coefficients of holomorphic cusp forms. These results are unconditional, but we demonstrate how to strengthen them under the Riemann Hypothesis. As an application, we show that there are infinitely many three term arithmetic progressions $n-h,n,n+h$ such that $a(n-h)a(n)a(n+h)\ne 0$.

17 pages (now with consistent author naming)

References in corpus (3)

• A note on the Laplace transform of the square in the circle problem
• A Shifted Sum for the Congruent Number Problem
• Non-real Poles and Irregularity of Distribution

Cited by in corpus (1)

• Arithmetic Progressions of Squares and Multiple Dirichlet Series