Mixed character sums modulo prime powers

TL;DR

This paper derives explicit bounds for mixed character sums modulo prime powers involving rational functions, extending understanding of their size and behavior in number theory.

Contribution

It provides new explicit estimates for mixed character sums with rational functions over prime power moduli, including bounds for both non-degenerate and degenerate cases.

Findings

Non-degenerate sums have bounds involving the degree D of the rational functions.

Bounds depend on the degrees of the numerator and denominator of the rational functions.

Explicit estimates are provided for sums involving rational functions over prime power moduli.

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.