Character sums of composite moduli and hybrid subconvexity

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$N$roughlyofsize$\sqrt M$and$W$ a smooth compactly supported weight function on $(0,\mathrm{\infty })$. As a corollary, we establish hybrid subconvexity bounds for the associated Dirichlet $L$-function.