Asymptotic bounds for special values of shifted convolution Dirichlet series

TL;DR

This paper investigates the asymptotic behavior of special values of shifted convolution Dirichlet series associated with cusp forms, using harmonic Maass forms to establish polynomial bounds as the shift parameter grows.

Contribution

It introduces a novel approach employing harmonic Maass forms to analyze the asymptotics of shifted convolution series, extending previous work on their Fourier coefficients.

Findings

Established polynomial bounds for the series as h approaches infinity

Connected the series' values to Fourier coefficients of mixed mock modular forms

Applied harmonic Maass form theory to a new class of L-series

Abstract

Hoffstein and Hulse defined the shifted convolution series of two cusp forms by "shifting" the usual Rankin-Selberg convolution L-series by a parameter h. We use the theory of harmonic Maass forms to study the behavior in h-aspect of certain values of these series and prove a polynomial bound as h approaches infinity. Our method relies on a result of Mertens and Ono, who showed that these values are Fourier coefficients of mixed mock modular forms.