We show that the binomial and related multiplicative character sums
have a simple evaluation for large enough (for if ).
On montre que les sommes binomiales et liées de caractères multiplicatifs
ont une évaluation simple pour suffisamment grand (pour si ).
Revised:
Accepted:
Published online:
DOI: 10.5802/jtnb.927
Mots-clés : Character Sums, Gauss sums, Jacobi Sums
Vincent Pigno 1; Christopher Pinner 2
@article{JTNB_2016__28_1_39_0, author = {Vincent Pigno and Christopher Pinner}, title = {Binomial {Character} {Sums} {Modulo} {Prime} {Powers}}, journal = {Journal de th\'eorie des nombres de Bordeaux}, pages = {39--53}, publisher = {Soci\'et\'e Arithm\'etique de Bordeaux}, volume = {28}, number = {1}, year = {2016}, doi = {10.5802/jtnb.927}, zbl = {1416.11130}, mrnumber = {3464610}, language = {en}, url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.927/} }
TY - JOUR AU - Vincent Pigno AU - Christopher Pinner TI - Binomial Character Sums Modulo Prime Powers JO - Journal de théorie des nombres de Bordeaux PY - 2016 SP - 39 EP - 53 VL - 28 IS - 1 PB - Société Arithmétique de Bordeaux UR - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.927/ DO - 10.5802/jtnb.927 LA - en ID - JTNB_2016__28_1_39_0 ER -
%0 Journal Article %A Vincent Pigno %A Christopher Pinner %T Binomial Character Sums Modulo Prime Powers %J Journal de théorie des nombres de Bordeaux %D 2016 %P 39-53 %V 28 %N 1 %I Société Arithmétique de Bordeaux %U https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.927/ %R 10.5802/jtnb.927 %G en %F JTNB_2016__28_1_39_0
Vincent Pigno; Christopher Pinner. Binomial Character Sums Modulo Prime Powers. Journal de théorie des nombres de Bordeaux, Volume 28 (2016) no. 1, pp. 39-53. doi : 10.5802/jtnb.927. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.927/
[1] B. C. Berndt, R. J. Evans & K. S. Williams, Gauss and Jacobi sums, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, Inc., New York, 1998, A Wiley-Interscience Publication, xii+583 pages. | Zbl
[2] F. N. Castro & C. J. Moreno, « Mixed exponential sums over finite fields », Proc. Amer. Math. Soc. 128 (2000), no. 9, p. 2529-2537. | DOI | Zbl
[3] T. Cochrane, « Exponential sums modulo prime powers », Acta Arith. 101 (2002), no. 2, p. 131-149. | DOI | MR | Zbl
[4] T. Cochrane & C. Pinner, « Using Stepanov’s method for exponential sums involving rational functions », J. Number Theory 116 (2006), no. 2, p. 270-292. | DOI | MR | Zbl
[5] T. Cochrane & Z. Zheng, « Pure and mixed exponential sums », Acta Arith. 91 (1999), no. 3, p. 249-278. | DOI | MR | Zbl
[6] —, « Exponential sums with rational function entries », Acta Arith. 95 (2000), no. 1, p. 67-95. | DOI | MR | Zbl
[7] —, « A survey on pure and mixed exponential sums modulo prime powers », in Number theory for the millennium, I (Urbana, IL, 2000), A K Peters, Natick, MA, 2002, p. 273-300. | Zbl
[8] K. Gong, W. Veys & D. Wan, « Power moments of Kloosterman sums », . | arXiv | DOI | MR | Zbl
[9] D. Han, « A hybrid mean value involving two-term exponential sums and polynomial character sums », Czechoslovak Math. J. 64(139) (2014), no. 1, p. 53-62. | DOI | MR | Zbl
[10] P. A. Leonard & K. S. Williams, « Evaluation of certain Jacobsthal sums », Boll. Un. Mat. Ital. B (5) 15 (1978), no. 3, p. 717-723. | Zbl
[11] R. Lidl & H. Niederreiter, Finite fields, second ed., Encyclopedia of Mathematics and its Applications, vol. 20, Cambridge University Press, Cambridge, 1997, With a foreword by P. M. Cohn, xiv+755 pages.
[12] L. J. Mordell, « On Salié’s sum », Glasgow Math. J. 14 (1973), p. 25-26. | DOI | Zbl
[13] V. Pigno & C. Pinner, « Twisted monomial Gauss sums modulo prime powers », Funct. Approx. Comment. Math. 51 (2014), no. 2, p. 285-301. | DOI | MR | Zbl
[14] H. Salié, « Über die Kloostermanschen Summen », Math. Z. 34 (1932), no. 1, p. 91-109. | DOI | Zbl
[15] J. Wang, « On the Jacobi sums modulo », J. Number Theory 39 (1991), no. 1, p. 50-64. | DOI | Zbl
[16] A. Weil, « On some exponential sums », Proc. Nat. Acad. Sci. U. S. A. 34 (1948), p. 204-207. | DOI | MR | Zbl
[17] K. S. Williams, « Note on Salié’s sum », Proc. Amer. Math. Soc. 30 (1971), p. 393-394. | DOI | Zbl
[18] —, « On Salié’s sum », J. Number Theory 3 (1971), p. 316-317. | DOI | Zbl
[19] W. Zhang & Z. Xu, « On the Dirichlet characters of polynomials in several variables », Acta Arith. 121 (2006), no. 2, p. 117-124. | DOI | MR | Zbl
[20] W. Zhang & W. Yao, « A note on the Dirichlet characters of polynomials », Acta Arith. 115 (2004), no. 3, p. 225-229. | DOI | MR | Zbl
[21] W. Zhang & Y. Yi, « On Dirichlet characters of polynomials », Bull. London Math. Soc. 34 (2002), no. 4, p. 469-473. | DOI | MR | Zbl
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