Triple Correlation Sums of Coefficients of Cusp Forms

2019/11/20 by Hulse, Thomas A., Kuan, Chan Ieong, Lowry-Duda, David +1
#11F11 #11M32 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1911.09216

Abstract

We produce nontrivial asymptotic estimates for shifted sums of the form ∑ 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) ≠ 0.

Related

• Triple correlations of Fourier coefficients of cusp forms
• Note on Triple correlations of Fourier coefficients of cusp forms
• Sums of Cusp Form Coefficients Along Quadratic Sequences
• On binary correlations of Fourier coefficients of holomorphic cusp forms at prime arguments
• A Note On Signs Of Fourier Coefficients Of Two Cusp Forms