Jacobi Sum Matrices

2012 ◽  
Vol 119 (2) ◽  
pp. 100
Author(s):  
Sam Vandervelde
Keyword(s):  
Jacobi Sum

scholarly journals On Jacobi sum Hecke characters ramified only at 2

1991 ◽  
Vol 38 (2) ◽  
pp. 161-184 ◽  
Author(s):  
Despina T. Prapavessi
Keyword(s):  
Jacobi Sum

EULER'S CRITERION FOR SEPTIC NONRESIDUES

2010 ◽  
Vol 06 (06) ◽  
pp. 1329-1347 ◽  
Author(s):  
JAGMOHAN TANTI ◽  
S. A. KATRE
Keyword(s):  
Explicit Expression ◽  
Root Of Unity ◽  
Jacobi Sum ◽  
Diophantine System ◽  
Mod P

Let p be a prime ≡ 1 (mod 7). In this paper, we obtain an explicit expression for a primitive seventh root of unity ( mod p) in terms of coefficients of a Jacobi sum of order 7 and also in terms of a solution of a Diophantine system of Leonard and Williams, and then obtain Euler's criterion for septic nonresidues D ( mod p) in terms of this seventh root. Explicit results are given for septic nonresidues for D = 2, 3, 5, 7.

scholarly journals On the distribution of Jacobi sums

2018 ◽  
Vol 2018 (741) ◽  
pp. 67-86
Author(s):  
Qing Lu ◽  
Weizhe Zheng ◽  
Zhiyong Zheng
Keyword(s):  
Finite Field ◽  
Unit Circle ◽  
Jacobi Sums ◽  
Jacobi Sum ◽  
Multiplicative Characters

Abstract Let \mathbf{F}_{q} be a finite field of q elements. For multiplicative characters \chi_{1},\ldots,\chi_{m} of \mathbf{F}_{q}^{\times} , we let J(\chi_{1},\ldots,\chi_{m}) denote the Jacobi sum. Nicholas Katz and Zhiyong Zheng showed that for m=2 , the normalized Jacobi sum q^{-1/2}J(\chi_{1},\chi_{2}) ( \chi_{1}\chi_{2} nontrivial) is asymptotically equidistributed on the unit circle as q\to\infty , when \chi_{1} and \chi_{2} run through all nontrivial multiplicative characters of \mathbf{F}_{q}^{\times} . In this paper, we show a similar property for m\geq 2 . More generally, we show that the normalized Jacobi sum q^{-(m-1)/2}J(\chi_{1},\ldots,\chi_{m}) ( \chi_{1}\cdots\chi_{m} nontrivial) is asymptotically equidistributed on the unit circle, when \chi_{1},\ldots,\chi_{m} run through arbitrary sets of nontrivial multiplicative characters of \mathbf{F}_{q}^{\times} with two of the sets being sufficiently large. The case m=2 answers a question of Shparlinski.

scholarly journals Evaluating prime power Gauss and Jacobi sums

2017 ◽  
Vol 48 (3) ◽  
pp. 227-240 ◽  
Author(s):  
Misty Ostergaard ◽  
Vincent Pigno ◽  
Christopher Pinner
Keyword(s):  
Prime Power ◽  
Gauss Sums ◽  
Jacobi Sums ◽  
Jacobi Sum ◽  
Simple Evaluation ◽  
Gauss And Jacobi Sums ◽  
Mod P

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 $p=2$.

Barnes' First Lemma and its Finite Analogue

1993 ◽  
Vol 36 (3) ◽  
pp. 273-282 ◽  
Author(s):  
Anna Helversen-Pasotto ◽  
Patrick Solé
Keyword(s):  
Gamma Function ◽  
Mellin Transform ◽  
Beta Function ◽  
Classical Case ◽  
Residue Theorem ◽  
Finite Case ◽  
Gaussian Sum ◽  
Jacobi Sum

AbstractWe give a parallel proof of Barnes' first lemma and of its finite analogue. In both cases we use the Mellin transform. In the classical case, the proof avoids the residue theorem. In the finite case the Gamma function is replaced by the Gaussian sum function and the beta function by the Jacobi sum function.

scholarly journals On Special Values of Jacobi-Sum HeckeL-Functions

2015 ◽  
Vol 24 (2) ◽  
pp. 247-259 ◽  
Author(s):  
Noriyuki Otsubo
Keyword(s):  
Special Values ◽  
Jacobi Sum

scholarly journals Note on the jacobi sum $J(\chi,\chi)$

1995 ◽  
Vol 7 (2) ◽  
pp. 461-471
Author(s):  
Stanislav Jakubec
Keyword(s):  
Jacobi Sum
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