A note on the moments of Kloosterman sums
HTML articles powered by AMS MathViewer
- by Ping Xi and Yuan Yi PDF
- Proc. Amer. Math. Soc. 141 (2013), 1233-1240 Request permission
Abstract:
In this note, we deduce an asymptotic formula for even power moments of Kloosterman sums based on the important work of N. M. Katz on Kloosterman sheaves. In a similar manner, we can also obtain an upper bound for odd power moments. Moreover, we shall give an asymptotic formula for odd power moments of absolute Kloosterman sums. Consequently, we find that there are infinitely many $a\bmod p$ such that $S(a,1;p)\gtrless 0$ as $p\rightarrow +\infty .$References
- H. Davenport, On certain exponential sums, J. Reine Angew. Math., 169 (1933), 158-176.
- Ronald Evans, Seventh power moments of Kloosterman sums, Israel J. Math. 175 (2010), 349–362. MR 2607549, DOI 10.1007/s11856-010-0014-0
- K. Hulek, J. Spandaw, B. van Geemen, and D. van Straten, The modularity of the Barth-Nieto quintic and its relatives, Adv. Geom. 1 (2001), no. 3, 263–289. MR 1874236, DOI 10.1515/advg.2001.017
- Henryk Iwaniec, Topics in classical automorphic forms, Graduate Studies in Mathematics, vol. 17, American Mathematical Society, Providence, RI, 1997. MR 1474964, DOI 10.1090/gsm/017
- Nicholas M. Katz, Sommes exponentielles, Astérisque, vol. 79, Société Mathématique de France, Paris, 1980 (French). Course taught at the University of Paris, Orsay, Fall 1979; With a preface by Luc Illusie; Notes written by Gérard Laumon; With an English summary. MR 617009
- H. D. Kloosterman, On the representation of numbers in the form $ax^2+by^2+cz^2+dt^2$, Acta Math., 49 (1926), 407-464.
- Chunlei Liu, Twisted higher moments of Kloosterman sums, Proc. Amer. Math. Soc. 130 (2002), no. 7, 1887–1892. MR 1896019, DOI 10.1090/S0002-9939-02-06510-3
- Ron Livné, Motivic orthogonal two-dimensional representations of $\textrm {Gal}(\overline \textbf {Q}/\mathbf Q)$, Israel J. Math. 92 (1995), no. 1-3, 149–156. MR 1357749, DOI 10.1007/BF02762074
- C. Peters, J. Top, and M. van der Vlugt, The Hasse zeta function of a $K3$ surface related to the number of words of weight $5$ in the Melas codes, J. Reine Angew. Math. 432 (1992), 151–176. MR 1184764
- H. Poincaré, Fonctions modulaires et fonctions fuchsiennes, Ann. Fac. Sci. Toulouse Sci. Math. Sci. Phys. (3) 3 (1911), 125–149 (French). MR 1508326, DOI 10.5802/afst.275
- John Riordan, Combinatorial identities, John Wiley & Sons, Inc., New York-London-Sydney, 1968. MR 0231725
- I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products, 7th ed., Elsevier/Academic Press, Amsterdam, 2007. Translated from the Russian; Translation edited and with a preface by Alan Jeffrey and Daniel Zwillinger; With one CD-ROM (Windows, Macintosh and UNIX). MR 2360010
- Hans Salié, Zur Abschätzung der Fourierkoeffizienten ganzer Modulformen, Math. Z. 36 (1933), no. 1, 263–278 (German). MR 1545344, DOI 10.1007/BF01188622
- Wolfgang M. Schmidt, Equations over finite fields. An elementary approach, Lecture Notes in Mathematics, Vol. 536, Springer-Verlag, Berlin-New York, 1976. MR 0429733, DOI 10.1007/BFb0080437
- D. I. Tolev, An identity for the Kloosterman sum, http://arxiv.org/abs/1007.2054.
- André Weil, On some exponential sums, Proc. Nat. Acad. Sci. U.S.A. 34 (1948), 204–207. MR 27006, DOI 10.1073/pnas.34.5.204
Additional Information
- Ping Xi
- Affiliation: School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an 710049, People’s Republic of China
- Email: pingxi.cn@gmail.com
- Yuan Yi
- Affiliation: School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an 710049, People’s Republic of China
- Email: yuanyi@mail.xjtu.edu.cn
- Received by editor(s): January 18, 2011
- Received by editor(s) in revised form: May 19, 2011, July 21, 2011, and August 22, 2011
- Published electronically: September 7, 2012
- Communicated by: Kathrin Bringmann
- © Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 141 (2013), 1233-1240
- MSC (2010): Primary 11L05
- DOI: https://doi.org/10.1090/S0002-9939-2012-11374-7
- MathSciNet review: 3008871