Binomial Character Sums Modulo Prime Powers

TL;DR

This paper provides explicit evaluations of binomial and multiplicative character sums modulo prime powers, extending understanding of their behavior for large exponents in number theory.

Contribution

It introduces simple evaluation formulas for binomial and related character sums modulo prime powers when the exponent is sufficiently large.

Findings

Explicit formulas for character sums when m ≥ 2

Simplification of sums for large prime power exponents

Enhanced understanding of multiplicative character sums

Abstract

We show that the binomial and related multiplicative character sums $$ \sum_{\stackrel{x=1}{(x,p)=1}}^{p^m} \chi (x^l(Ax^k +B)^w),\hspace{3ex} \sum_{x=1}^{p^m} \chi_1 (x)\chi_2(Ax^k +B), $$ have a simple evaluation for large enough $m$ (for $m\geq 2$ if $p\nmid ABk$).