A note on signs of Kloosterman sums

Kaisa Matomäki

Bulletin de la Société Mathématique de France (2011)

  • Volume: 139, Issue: 3, page 287-295
  • ISSN: 0037-9484

Abstract

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We prove that the sign of Kloosterman sums Kl ( 1 , 1 ; n ) changes infinitely often as n runs through the square-free numbers with at most 15 prime factors. This improves on a previous result by Sivak-Fischler who obtained 18 instead of 15. Our improvement comes from introducing an elementary inequality which gives lower and upper bounds for the dot product of two sequences whose individual distributions are known.

How to cite

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Matomäki, Kaisa. "A note on signs of Kloosterman sums." Bulletin de la Société Mathématique de France 139.3 (2011): 287-295. <http://eudml.org/doc/272621>.

@article{Matomäki2011,
abstract = {We prove that the sign of Kloosterman sums $\operatorname\{Kl\}(1, 1; n)$ changes infinitely often as $n$ runs through the square-free numbers with at most $15$ prime factors. This improves on a previous result by Sivak-Fischler who obtained 18 instead of 15. Our improvement comes from introducing an elementary inequality which gives lower and upper bounds for the dot product of two sequences whose individual distributions are known.},
author = {Matomäki, Kaisa},
journal = {Bulletin de la Société Mathématique de France},
keywords = {Kloosterman sums; rearrangement inequality; Sato-Tate conjecture},
language = {eng},
number = {3},
pages = {287-295},
publisher = {Société mathématique de France},
title = {A note on signs of Kloosterman sums},
url = {http://eudml.org/doc/272621},
volume = {139},
year = {2011},
}

TY - JOUR
AU - Matomäki, Kaisa
TI - A note on signs of Kloosterman sums
JO - Bulletin de la Société Mathématique de France
PY - 2011
PB - Société mathématique de France
VL - 139
IS - 3
SP - 287
EP - 295
AB - We prove that the sign of Kloosterman sums $\operatorname{Kl}(1, 1; n)$ changes infinitely often as $n$ runs through the square-free numbers with at most $15$ prime factors. This improves on a previous result by Sivak-Fischler who obtained 18 instead of 15. Our improvement comes from introducing an elementary inequality which gives lower and upper bounds for the dot product of two sequences whose individual distributions are known.
LA - eng
KW - Kloosterman sums; rearrangement inequality; Sato-Tate conjecture
UR - http://eudml.org/doc/272621
ER -

References

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  1. [1] T. Estermann – « On Kloosterman’s sum », Mathematika8 (1961), p. 83–86. Zbl0114.26302MR126420
  2. [2] E. Fouvry & P. Michel – « Crible asymptotique et sommes de Kloosterman », in Proceedings of the Session in Analytic Number Theory and Diophantine Equations, Bonner Mathematische Schriften, vol. 360, Univ. Bonn, 2003. Zbl1065.11075MR2075623
  3. [3] —, « Sur le changement de signe des sommes de Kloosterman », Annals of Mathematics165 (2007), p. 675–715. Zbl1230.11096MR2335794
  4. [4] G. H. Hardy, J. E. Littlewood & G. Pólya – Inequalities, Cambridge Univ. Press, Cambridge, 2001, reprint of the 1954 Second Edition. Zbl0010.10703
  5. [5] N. M. Katz – Gauss sums, Kloosterman sums, and monodromy groups, Princeton Univ. Press, Princeton, 1988. Zbl0675.14004MR955052
  6. [6] J. Sivak-Fischler – « Crible étrange et sommes de Kloosterman », Acta Arith.128 (2007), p. 69–100. Zbl1230.11097MR2306565
  7. [7] —, « Crible asymptotique et sommes de Kloosterman », Bull. Soc. Math. France137 (2009), p. 1–62. Zbl1258.11079MR2496700

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