Mixed character sums modulo prime powers

Abstract

We obtain explicit estimates for the mixed character sum $S=S(\ddot{I}‡,g,f,p^m)={\sum }_{x=1}^{p^m}\ddot{I}‡(g(x))e_{p^m}(f(x))$, where $p^m$ is a prime power, $\ddot{I}‡$ is a multiplicative character mod $p^m$ and $f,g$ are rational functions over $\mathbb{Q}$. Let $f=f_{+}/f_{-}$, $g=g_{+}/g_{-}$ in reduced form, and set $D=\text{deg}(f)+Z-1$ where $Z$ is the number of distinct complex zeros of $f_{-}{g}_{+}{g}_{-}$, and $\hat{I}”=\text{deg}(f)+\text{deg}(g)$ for polynomial $f,g$, $\hat{I}”=2(\text{deg}(f)+\text{deg}(g))$ otherwise. We show for example that for odd $p$, any non-degenerate sum has $|S|\le 3^{4/3}{p}^{m(1-\frac{1}{D})}$ if ${\text{deg}}_p(f)\ge 1$, and $|S|\le 3^{4/3}{p}^{m(1-\frac{1}{\hat{I}}”)}$ if ${\text{deg}}_p(g)\ge 1$. Analogous bounds are given for degenerate sums.