Evaluating prime power Gauss and Jacobi sums

Misty Ostergaard, Vincent Pigno, Christopher Pinner


We show that for any mod $p^m$ characters, $\chi_1, \dots, \chi_k,$ with at least one $\chi_i$ primitive mod $p^m$, the Jacobi sum, $$ \mathop{\sum_{x_1=1}^{p^m}\dots \sum_{x_k=1}^{p^m}}_{x_1+\dots+x_k\equiv B \text{ mod } p^m}\chi_1(x_1)\cdots \chi_k(x_k), $$ has a simple evaluation when $m$ is sufficiently large (for $m\geq 2$ if $p\nmid B$). As part of the proof we give a simple evaluation of the mod $p^m$ Gauss sums when $m\geq 2$ that differs slightly from existing evaluations when $p=2$.


Gauss Sums; Jacobi Sums; Character Sums; Exponential Sums

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DOI: http://dx.doi.org/10.5556/j.tkjm.48.2017.1866

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