Character sums of composite moduli and hybrid subconvexity

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.