# A note on signs of Kloosterman sums

Bulletin de la Société Mathématique de France (2011)

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

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topMatomäki, 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

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

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