Character sums of composite moduli and hybrid subconvexity

TL;DR

This paper develops a new method using the $\delta$-symbol to achieve cancellation in smooth character sums for composite moduli, leading to hybrid subconvexity bounds for related Dirichlet L-functions.

Contribution

It introduces a $\delta$-symbol approach to obtain non-trivial bounds for character sums with composite moduli, advancing subconvexity results.

Findings

Established non-trivial bounds for smooth character sums of size roughly $\sqrt{M}$

Derived hybrid subconvexity bounds for Dirichlet L-functions with composite moduli

Provided a novel $\delta$-symbol method applicable to character sums

Abstract

Let $M=M_1 M_2 M_3$ be the product of three distinct primes and let $\chi=\chi_1 \chi_2 \chi_3$ be a Dirichlet character of modulus $M$ such that each $\chi_i$ is a primitive character modulo $M_i$ for $i=1,2,3$. In this paper, we provide a $\delta$-symbol method for obtaining non-trivial cancellation in smooth character sums of the form $\sum_{n=1}^\infty \chi(n) W(n/N)$, with $N$ roughly of size $\sqrt M$ and $W$ a smooth compactly supported weight function on $(0, \infty)$. As a corollary, we establish hybrid subconvexity bounds for the associated Dirichlet $L$-function.