Binomial Character Sums Modulo Prime Powers

Abstract

We show that the binomial and related multiplicative character sums $$\sum _{\stackrel{x=1}{(x,p)=1}}^{p^m}\chi (x^l(A{x}^k+B{)}^w),\sum _{x=1}^{p^m}{\chi }_1(x){\chi }_2(A{x}^k+B),$$ have a simple evaluation for large enough $m$ (for $m\ge 2$ if $p\nmid ABk$).