Rankin-Selberg coefficients in large arithmetic progressions

Abstract

Let $({\lambda }_f(n){)}_{n\ge 1}$ be the Hecke eigenvalues of either a holomorphic Hecke eigencuspform or a Hecke-Maass cusp form $f$. We prove that, for any fixed $\eta >0$, under the Ramanujan-Petersson conjecture for $\mathrm{G}{\mathrm{L}}_2$ Maass forms, the Rankin-Selberg coefficients $({\lambda }_f(n{)}^2{)}_{n\ge 1}$ admit a level of distribution $\theta =2/5+1/260-\eta$ in arithmetic progressions.

accepted for publication in SCIENCE CHINA Mathematics