Evaluating prime power Gauss and Jacobi sums

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Published: Sep 30, 2017
DOI: https://doi.org/10.5556/j.tkjm.48.2017.1866
Keywords:
Gauss Sums Jacobi Sums Character Sums Exponential Sums

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Misty Ostergaard
Vincent Pigno
Christopher Pinner

Abstract

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 $p=2$.

Article Details

How to Cite
Ostergaard, M., Pigno, V., & Pinner, C. (2017). Evaluating prime power Gauss and Jacobi sums. Tamkang Journal of Mathematics, 48(3), 227–240. https://doi.org/10.5556/j.tkjm.48.2017.1866
Issue
Vol. 48 No. 3 (2017)
Section
Papers
Author Biographies

Misty Ostergaard

Department ofMathematics, University of Southern Indiana, Evansville, IN 47712.

Vincent Pigno

Department ofMathematics & Statistics, California State University, Sacramento, Sacramento, CA 95819.

Christopher Pinner

Department ofMathematics, Kansas State University,Manhattan, KS 66506.

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