Mixed character sums modulo prime powers

Abstract

We obtain explicit estimates for the mixed character sum $S= S(\chi,g,f,p^m) = \sum_{x=1}^{p^m} \chi (g(x)) e_{p^m}(f(x))$, where $p^m$ is a prime power, $\chi$ 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 $\Delta= \text{deg}(f)+\text{deg}(g)$ for polynomial $f,g$, $\Delta=2(\text{deg}(f)+\text{deg}(g))$ otherwise. We show for example that for odd $p$, any non-degenerate sum has $|S|\le 3^{4/3}\, p^{m(1-\frac 1D)}$ if $\text{deg}_p(f) \ge 1$, and $|S| \le 3^{4/3}\, p^{m(1-\frac 1\Delta)}$ if $\text{deg}_p(g) \ge 1$. Analogous bounds are given for degenerate sums.