Bounds for short character sums for $GL(2) \times GL(3)$ twists

TL;DR

This paper establishes new bounds for short character sums involving automorphic forms on $GL(2)$ and $GL(3)$, advancing understanding of their behavior in analytic number theory.

Contribution

It provides novel bounds for twisted character sums for $GL(2) imes GL(3)$ automorphic forms, extending previous results in the field.

Findings

Bound for $S_{ ext{pi,f,chi}}(N)$: $oxed{N^{3/4}p^{11/16 + ext{eta}/4}(Np)^ ext{epsilon}}$

Improved estimates for character sums involving automorphic forms on $GL(2)$ and $GL(3)$

Enhanced understanding of the distribution of automorphic forms in short intervals

Abstract

Let $\pi$ be a $SL(3,\mathbb{Z})$ Hecke Maass-cusp form, $f$ be a $SL(2,\mathbb{Z})$ holomorphic cusp form or Maass-cusp form with normalized Fourier coefficients $\lambda_{\pi}(r,n) \text{ and }\lambda_{f}(n)$ respectively and $\chi$ be any non-trivial character mod $p$ where $p$ is a prime. Then we have $$S_{\pi ,f,\chi }(N)\ll_{\pi , f, \epsilon} N^{3/4}p^{11/16 +{\eta /4}}(Np)^\epsilon .$$