On triple correlations of Fourier coefficients of cusp forms

Abstract

Assume that α,β are arbitrary arithmetic functions, and that γ is an arithmetic function, which does not correlate with additive characters. By introducing a simple argument, we are able to give a general upper bound for the triple correlation∑n∈(X,2X]α(n)β(n−h)γ(n+h) averaging over a short variable h. More precise estimates can be obtained by taking γ to be Fourier coefficients of cusp forms, which improve and streamline the results of Lin and Singh.