Triple Correlation Sums of Coefficients of Cusp Forms

TL;DR

This paper develops asymptotic estimates for shifted sums of Fourier coefficients of cusp forms, demonstrating their non-vanishing in infinitely many three-term arithmetic progressions, with results strengthened under the Riemann Hypothesis.

Contribution

It provides new unconditional asymptotic estimates for shifted sums of cusp form coefficients and shows their application to non-vanishing in arithmetic progressions, with potential improvements assuming RH.

Findings

Unconditional asymptotic estimates for shifted sums of cusp form coefficients.

Existence of infinitely many three-term arithmetic progressions with non-zero coefficients.

Strengthening of results under the Riemann Hypothesis.

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 $a(n-h)a(n)a(n+h) \neq 0$.