Mixed character sums modulo prime powers
Todd Cochrane, Andrew Granville

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 , where is a prime power, is a multiplicative character mod and are rational functions over . Let , in reduced form, and set where is the number of distinct complex zeros of , and for polynomial , otherwise. We show for example that for odd , any non-degenerate sum has if , and if . Analogous bounds are given for degenerate sums.
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