Sign changes of Kloosterman sums and exceptional characters

Drappeau, Sary; Maynard, James

doi:10.1090/proc/14239

Sign changes of Kloosterman sums and exceptional characters

Authors: Sary Drappeau and James Maynard
Journal: Proc. Amer. Math. Soc. 147 (2019), 61-75
MSC (2010): Primary 11L05, 11N36; Secondary 11N75, 11L20, 11M20
DOI: https://doi.org/10.1090/proc/14239
Published electronically: October 3, 2018
MathSciNet review: 3876731
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We prove that the existence of exceptional real zeros of Dirichlet -functions would lead to cancellations in the sum $\sum _{p\leq x} \textup {Kl}(1, p)$ of Kloosterman sums over primes, and also to sign changes of $\textup {Kl}(1, n)$ , where runs over integers with exactly two prime factors. Our arguments involve a variant of Bombieri's sieve, bounds for twisted sums of Kloosterman sums, and work of Fouvry and Michel on sums of $\left \vert \textup {Kl}(1, n)\right \vert$ .

[BM15] Valentin Blomer and Djordje Milićević, Kloosterman sums in residue classes, J. Eur. Math. Soc. (JEMS) 17 (2015), no. 1, 51–69. MR 3312403, https://doi.org/10.4171/JEMS/498
[Bom76] Enrico Bombieri, On twin almost primes, Acta Arith. 28 (1975/76), no. 2, 177–193. MR 396435, https://doi.org/10.4064/aa-28-2-177-193
[DI83] J.-M. Deshouillers and H. Iwaniec, Kloosterman sums and Fourier coefficients of cusp forms, Invent. Math. 70 (1982/83), no. 2, 219–288. MR 684172, https://doi.org/10.1007/BF01390728
[Dra17] Sary Drappeau, Sums of Kloosterman sums in arithmetic progressions, and the error term in the dispersion method, Proc. Lond. Math. Soc. (3) 114 (2017), no. 4, 684–732. MR 3653244, https://doi.org/10.1112/plms.12022
[FKM14] Étienne Fouvry, Emmanuel Kowalski, and Philippe Michel, Algebraic trace functions over the primes, Duke Math. J. 163 (2014), no. 9, 1683–1736. MR 3217765, https://doi.org/10.1215/00127094-2690587
[FM03a] E. Fouvry and P. Michel, Crible asymptotique et sommes de Kloosterman, Proceedings of the Session in Analytic Number Theory and Diophantine Equations, Bonner Math. Schriften, vol. 360, Univ. Bonn, Bonn, 2003, pp. 27 (French). MR 2075623
[FM03b] Etienne Fouvry and Philippe Michel, Sommes de modules de sommes d’exponentielles, Pacific J. Math. 209 (2003), no. 2, 261–288 (French, with English summary). MR 1978371, https://doi.org/10.2140/pjm.2003.209.261
[FM06] Étienne Fouvry and Philippe Michel, Errata to the article: ``Sums of moduli of exponential sums'' (French) [Pacific J. Math. 209 (2003), no. 2, 261-288; mr1978371], Pacific J. Math. 225 (2006), no. 1, 199-200.
[FM07] É. Fouvry and Ph. Michel, Sur le changement de signe des sommes de Kloosterman, Ann. of Math. (2) 165 (2007), no. 3, 675–715 (French, with English summary). MR 2335794, https://doi.org/10.4007/annals.2007.165.675
[FI03] John B. Friedlander and Henryk Iwaniec, Exceptional characters and prime numbers in arithmetic progressions, Int. Math. Res. Not. 37 (2003), 2033–2050. MR 1995146, https://doi.org/10.1155/S1073792803130784
[FI04] John B. Friedlander and Henryk Iwaniec, Exceptional characters and prime numbers in short intervals, Selecta Math. (N.S.) 10 (2004), no. 1, 61–69. MR 2061223, https://doi.org/10.1007/s00029-004-0374-6
[FI05] J. B. Friedlander and H. Iwaniec, The illusory sieve, Int. J. Number Theory 1 (2005), no. 4, 459–494. MR 2196790, https://doi.org/10.1142/S1793042105000303
[FI10] John Friedlander and Henryk Iwaniec, Opera de cribro, American Mathematical Society Colloquium Publications, vol. 57, American Mathematical Society, Providence, RI, 2010. MR 2647984
[FI13] J. B. Friedlander and H. Iwaniec, Exceptional discriminants are the sum of a square and a prime, Q. J. Math. 64 (2013), no. 4, 1099–1107. MR 3151606, https://doi.org/10.1093/qmath/has038
[HB83] D. R. Heath-Brown, Prime twins and Siegel zeros, Proc. London Math. Soc. (3) 47 (1983), no. 2, 193–224. MR 703977, https://doi.org/10.1112/plms/s3-47.2.193
[Hoo64] C. Hooley, On the distribution of the roots of polynomial congruences, Mathematika 11 (1964), 39–49. MR 163874, https://doi.org/10.1112/S0025579300003466
[Kat88] Nicholas M. Katz, Gauss sums, Kloosterman sums, and monodromy groups, Annals of Mathematics Studies, vol. 116, Princeton University Press, Princeton, NJ, 1988. MR 955052
[Kim03] Henry H. Kim, Functoriality for the exterior square of 𝐺𝐿₄ and the symmetric fourth of 𝐺𝐿₂, J. Amer. Math. Soc. 16 (2003), no. 1, 139–183. With appendix 1 by Dinakar Ramakrishnan and appendix 2 by Kim and Peter Sarnak. MR 1937203, https://doi.org/10.1090/S0894-0347-02-00410-1
[Klo27] H. D. Kloosterman, On the representation of numbers in the form 𝑎𝑥²+𝑏𝑦²+𝑐𝑧²+𝑑𝑡², Acta Math. 49 (1927), no. 3-4, 407–464. MR 1555249, https://doi.org/10.1007/BF02564120
[Kuz80] N. V. Kuznecov, The Petersson conjecture for cusp forms of weight zero and the Linnik conjecture. Sums of Kloosterman sums, Mat. Sb. (N.S.) 111(153) (1980), no. 3, 334–383, 479 (Russian). MR 568983
[LRS95] W. Luo, Z. Rudnick, and P. Sarnak, On Selberg’s eigenvalue conjecture, Geom. Funct. Anal. 5 (1995), no. 2, 387–401. MR 1334872, https://doi.org/10.1007/BF01895672
[Mat11] Kaisa Matomäki, A note on signs of Kloosterman sums, Bull. Soc. Math. France 139 (2011), no. 3, 287–295 (English, with English and French summaries). MR 2869308, https://doi.org/10.24033/bsmf.2609
[Poi11] H. Poincaré, Fonctions modulaires et fonctions fuchsiennes, Ann. Fac. Sci. Toulouse Sci. Math. Sci. Phys. (3) 3 (1911), 125–149 (French). MR 1508326
[Sie35] C. L. Siegel, Über die Classenzahl quadratischer Zahlkörper., Acta Arith. 1 (1935), 83-86.
[SF07] Jimena Sivak-Fischler, Crible étrange et sommes de Kloosterman, Acta Arith. 128 (2007), no. 1, 69–100 (French). MR 2306565, https://doi.org/10.4064/aa128-1-4
[SF09] Jimena Sivak-Fischler, Crible asymptotique et sommes de Kloosterman, Bull. Soc. Math. France 137 (2009), no. 1, 1–62 (French, with English and French summaries). MR 2496700, https://doi.org/10.24033/bsmf.2568
[Top15] Berke Topacogullari, On a certain additive divisor problem, Acta Arith. 181 (2017), no. 2, 143–172. MR 3726186, https://doi.org/10.4064/aa8643-5-2017
[Wei48] André Weil, On some exponential sums, Proc. Nat. Acad. Sci. U.S.A. 34 (1948), 204–207. MR 27006, https://doi.org/10.1073/pnas.34.5.204
[Xi15a] Ping Xi, Sign changes of Kloosterman sums with almost prime moduli, Monatsh. Math. 177 (2015), no. 1, 141–163. MR 3336337, https://doi.org/10.1007/s00605-014-0653-z
[Xi15b] Ping Xi, Sign changes of Kloosterman sums with almost prime moduli. II, Int. Math. Res. Not. IMRN 4 (2018), 1200–1227. MR 3801460, https://doi.org/10.1093/imrn/rnw276
[Xi18] Ping Xi, When Kloosterman sums meet Hecke eigenvalues, preprint (2018), arXiv, http://arxiv.org/abs/1801.07658.v2.