On the shifted convolution problem in mean

TL;DR

This paper investigates the mean value of the shifted convolution problem for Hecke eigenvalues of cusp forms, providing new bounds and extending results to weighted cases and holomorphic cusp forms.

Contribution

It introduces new upper bounds for the shifted convolution problem in mean for both non-holomorphic and holomorphic cusp forms, including weighted cases.

Findings

Established bounds for the shifted convolution problem over Hecke eigenvalues.

Extended results to weighted cases of the problem.

Provided analogous bounds for Fourier coefficients of holomorphic cusp forms.

Abstract

We study a mean value of the shifted convolution problem over the Hecke eigenvalues of a fixed non-holomorphic cusp form. We attain a result also for a weighted case. Furthermore, we point out that the proof yields analogous upper bounds for the shifted convolution problem over the Fourier coefficients of a fixed holomorphic cusp form in mean.