Exponential sums with multiplicative coefficients and applications
HTML articles powered by AMS MathViewer
- by Régis de la Bretèche and Andrew Granville PDF
- Trans. Amer. Math. Soc. 375 (2022), 6875-6901 Request permission
Abstract:
We show that if an exponential sum with multiplicative coefficients is large then the associated multiplicative function is “pretentious”. This leads to applications in the circle method, and a natural interpretation of the local-global principle.References
- Gennady Bachman, On exponential sums with multiplicative coefficients. II, Acta Arith. 106 (2003), no. 1, 41–57. MR 1956974, DOI 10.4064/aa106-1-3
- R. de la Bretèche, Sommes d’exponentielles et entiers sans grand facteur premier, Proc. London Math. Soc. (3) 77 (1998), no. 1, 39–78 (French). MR 1625487, DOI 10.1112/S0024611598000409
- R. de la Bretèche, Sommes sans grand facteur premier, Acta Arith. 88 (1999), no. 1, 1–14 (French). MR 1698349, DOI 10.4064/aa-88-1-1-14
- Jörg Brüdern, Binary additive problems and the circle method, multiplicative sequences and convergent sieves, Analytic number theory, Cambridge Univ. Press, Cambridge, 2009, pp. 91–132. MR 2508639
- Todd Cochrane, Exponential sums modulo prime powers, Acta Arith. 101 (2002), no. 2, 131–149. MR 1880304, DOI 10.4064/aa101-2-5
- Hédi Daboussi and Hubert Delange, Quelques propriétés des fonctions multiplicatives de module au plus égal à $1$, C. R. Acad. Sci. Paris Sér. A 278 (1974), 657–660 (French). MR 332702
- H. Davenport, Analytic methods for Diophantine equations and Diophantine inequalities, 2nd ed., Cambridge Mathematical Library, Cambridge University Press, Cambridge, 2005. With a foreword by R. C. Vaughan, D. R. Heath-Brown and D. E. Freeman; Edited and prepared for publication by T. D. Browning. MR 2152164, DOI 10.1017/CBO9780511542893
- P. D. T. A. Elliott, Extrapolating the mean-values of multiplicative functions, Nederl. Akad. Wetensch. Indag. Math. 51 (1989), no. 4, 409–420. MR 1041494, DOI 10.1016/1385-7258(89)90004-8
- P. D. T. A. Elliott, Multiplicative functions on arithmetic progressions. VII. Large moduli, J. London Math. Soc. (2) 66 (2002), no. 1, 14–28. MR 1911217, DOI 10.1112/S0024610702003228
- Ke Gong and ChaoHua Jia, Kloosterman sums with multiplicative coefficients, Sci. China Math. 59 (2016), no. 4, 653–660. MR 3474493, DOI 10.1007/s11425-015-5108-z
- Ke Gong and Chaohua Jia, Shifted character sums with multiplicative coefficients, J. Number Theory 153 (2015), 364–371. MR 3327581, DOI 10.1016/j.jnt.2015.01.015
- K. Gong, C. Jia, and M. A. Korolev, Shifted character sums with multiplicative coefficients, II, J. Number Theory 178 (2017), 31–39. MR 3646825, DOI 10.1016/j.jnt.2017.02.006
- Andrew Granville and K. Soundararajan, The spectrum of multiplicative functions, Ann. of Math. (2) 153 (2001), no. 2, 407–470. MR 1829755, DOI 10.2307/2661346
- Andrew Granville, Adam J. Harper, and K. Soundararajan, A new proof of Halász’s theorem, and its consequences, Compos. Math. 155 (2019), no. 1, 126–163. MR 3880027, DOI 10.1112/s0010437x18007522
- A. Granville and K. Soundararajan, Multiplicative number theory: An alternative approach, to appear.
- G. Halász, Über die Mittelwerte multiplikativer zahlentheoretischer Funktionen, Acta Math. Acad. Sci. Hungar. 19 (1968), 365–403 (German). MR 230694, DOI 10.1007/BF01894515
- Oleksiy Klurman, Correlations of multiplicative functions and applications, Compos. Math. 153 (2017), no. 8, 1622–1657. MR 3705270, DOI 10.1112/S0010437X17007163
- M. A. Korolëv, Kloosterman sums with multiplicative coefficients, Izv. Ross. Akad. Nauk Ser. Mat. 82 (2018), no. 4, 3–17 (Russian, with Russian summary); English transl., Izv. Math. 82 (2018), no. 4, 647–661. MR 3833472, DOI 10.4213/im8633
- W.-C. W. Li, Number theory with applications, World Scientific, Singapore, 1996.
- Wen-Ching Winnie Li, Character sums over $p$-adic fields, J. Number Theory 74 (1999), no. 2, 181–229. MR 1671665, DOI 10.1006/jnth.1998.2328
- Lilian Matthiesen, Linear correlations of multiplicative functions, Proc. Lond. Math. Soc. (3) 121 (2020), no. 2, 372–425. MR 4093960, DOI 10.1112/plms.12309
- H. L. Montgomery and R. C. Vaughan, Exponential sums with multiplicative coefficients, Invent. Math. 43 (1977), no. 1, 69–82. MR 457371, DOI 10.1007/BF01390204
- Gérald Tenenbaum, Introduction to analytic and probabilistic number theory, 3rd ed., Graduate Studies in Mathematics, vol. 163, American Mathematical Society, Providence, RI, 2015. Translated from the 2008 French edition by Patrick D. F. Ion. MR 3363366, DOI 10.1090/gsm/163
- E. C. Titchmarsh, The theory of the Riemann zeta-function, 2nd ed., The Clarendon Press, Oxford University Press, New York, 1986. Edited and with a preface by D. R. Heath-Brown. MR 882550
Additional Information
- Régis de la Bretèche
- Affiliation: Institut de Mathématiques de Jussieu-Paris Rive Gauche, Université Paris Cité, Sorbonne Université, CNRS UMR 7586, Case Postale 7012, F-75251 Paris CEDEX 13, France
- ORCID: 0000-0002-6713-9766
- Email: regis.delabreteche@imj-prg.fr
- Andrew Granville
- Affiliation: Départment de Mathématiques et Statistique, Université de Montréal, CP 6128 succ Centre-Ville, Montréal, Quebec H3C 3J7, Canada; and Department of Mathematics, University College London, Gower Street, London WC1E 6BT, England.
- MR Author ID: 76180
- ORCID: 0000-0001-8088-1247
- Email: andrew.granville@umontreal.ca
- Received by editor(s): March 31, 2021
- Received by editor(s) in revised form: November 12, 2021
- Published electronically: July 25, 2022
- Additional Notes: A.G. est partiellement soutenu par une bourse de la Conseil de recherches en sciences naturelles et en génie du Canada, and was also partially supported by a European Research Council grant, agreement n$^{\text {o}}$ 670239. Part of this work was undertaken while the second author was in Paris; his stay was supported by Université Paris–Diderot, Université Pierre et Marie Curie, Université d’Orsay and Fondation Sciences Mathématiques de Paris. The hospitality and financial support of these institutions is gratefully acknowledged.
- © Copyright 2022 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 375 (2022), 6875-6901
- DOI: https://doi.org/10.1090/tran/8625