Rankin--Selberg coefficients in arithmetic progressions modulo prime powers

TL;DR

This paper proves a new level of distribution for Rankin--Selberg coefficients in arithmetic progressions modulo prime powers, assuming the Ramanujan--Petersson conjecture for certain automorphic forms.

Contribution

It establishes a specific level of distribution for Rankin--Selberg coefficients in arithmetic progressions modulo prime powers, advancing understanding under conjectural assumptions.

Findings

Level of distribution $ heta=2/5+3/305- ext{epsilon}$ for coefficients in arithmetic progressions

Results hold for prime power moduli $q=p^k$, $k extgreater=2$, $p eq 3$

Assumes Ramanujan--Petersson conjecture for $ ext{GL}_2$ Maass forms

Abstract

Let $\varepsilon>0$ be given. For prime power moduli $q=p^k$ with $k\geq 2$ and $p\neq 3$, and assuming the Ramanujan--Petersson conjecture for $\GL_2$ Maass forms, we prove that the Rankin--Selberg coefficients $\{\lambda_f(n)^2\}_{n\geq 1}$ have a level of distribution $\theta=2/5+3/305-\varepsilon$ in arithmetic progressions $n \equiv a \bmod q$.