On the Rankin-Selberg problem in families

TL;DR

This paper studies the Rankin-Selberg problem within families of modular forms and Maass forms, providing new insights under certain average bounds and unconditionally for Maass forms.

Contribution

It offers novel results on the Rankin-Selberg problem in families, assuming Lindelöf-on-average bounds for holomorphic forms and without assumptions for Maass forms.

Findings

Results on the distribution of Rankin-Selberg convolutions in short intervals

Conditional bounds for holomorphic modular forms

Unconditional bounds for Maass cusp forms

Abstract

In this paper, we investigate the Rankin-Selberg problem over short intervals in families of holomorphic modular forms and Hecke-Maass cusp forms. Our investigation assumes a Lindel\"of-on-average bound for holomorphic modular forms, and for Hecke-Maass cusp forms, we make no assumptions.