Hostname: page-component-74656f9884-npj9w Total loading time: 0 Render date: 2026-11-09T09:00:22.135Z Has data issue: false hasContentIssue false
  • English
  • Français

Pretentious multiplicative functions and the prime number theorem for arithmetic progressions

Part of: Multiplicative number theory Zeta and $L$-functions: analytic theory

Published online by Cambridge University Press:  14 May 2013

Dimitris Koukoulopoulos
Dimitris Koukoulopoulos*
Affiliation:
Département de mathématiques et de statistique, Université de Montréal, CP 6128 succ. Centre-Ville, Montréal, QC H3C 3J7, Canada email koukoulo@dms.umontreal.ca
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

Building on the concept of pretentious multiplicative functions, we give a new and largely elementary proof of the best result known on the counting function of primes in arithmetic progressions.

Keywords

prime number theorem for arithmetic progressionsmultiplicative functionsHalász’s theoremelementary proofs of the prime number theoremSiegel’s theorem

MSC classification

Secondary: 11M06: $zeta (s)$ and $L(s, chi)$ 11M20: Real zeros of $L(s, chi)$; results on $L(1, chi)$ 11N05: Distribution of primes 11N13: Primes in progressions

Information

Type
Research Article
Information
Compositio Mathematica , Volume 149 , Issue 7 , July 2013 , pp. 1129 - 1149
Copyright
© The Author(s) 2013 

References

Bombieri, E., Maggiorazione del resto nel ‘Primzahlsatz’ col metodo di Erdős–Selberg, Ist. Lombardo Accad. Sci. Lett. Rend. A 96 (1962), 343–350 (in Italian).Google Scholar
Daboussi, H., Sur le théorème des nombres premiers. [On the prime number theorem], C. R. Acad. Sci. Paris Sér. I Math. 298 (1984), 161–164 (in French, with a summary in English).Google Scholar
Davenport, H., Multiplicative number theory, Graduate Texts in Mathematics, vol. 74, third edition (Springer, New York, NY, 2000), revised and with a preface by Hugh L. Montgomery.Google Scholar
Diamond, H. G. and Steinig, J., An elementary proof of the prime number theorem with a remainder term, Invent. Math. 11 (1970), 199–258.CrossRefGoogle Scholar
Elliott, P. D. T. A., Probabilistic number theory. I. Mean-value theorems, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Science], vol. 239 (Springer, Berlin, 1979).Google Scholar
Erdős, P., On a new method in elementary number theory which leads to an elementary proof of the prime number theorem, Proc. Natl. Acad. Sci. USA 35 (1949), 374–384.Google Scholar
Ford, K., Vinogradov’s integral and bounds for the Riemann zeta function, Proc. Lond. Math. Soc. (3) 85 (2002), 565–633.CrossRefGoogle Scholar
Friedlander, J. and Iwaniec, H., On Bombieri’s asymptotic sieve, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 5 (1978), 719–756.Google Scholar
Goldfeld, D. M., The elementary proof of the prime number theorem: an historical perspective, in Number theory (New York, 2003) (Springer, New York, NY, 2004), 179–192.Google Scholar
Granville, A., Different approaches to the distribution of prime numbers, Milan J. Math. 78 (2009), 1–25.Google Scholar
Granville, A. and Soundararajan, K., Decay of mean values of multiplicative functions, Canad. J. Math. 55 (2003), 1191–1230.Google Scholar
Granville, A. and Soundararajan, K., Pretentious multiplicative functions and an inequality for the zeta-function, in Anatomy of integers, CRM Proceedings & Lecture Notes, vol. 46 (American Mathematical Society, Providence, RI, 2008), 191–197.Google Scholar
Granville, A. and Soundararajan, K., Multiplicative number theory: the pretentious approach, to appear.Google Scholar
Halász, G., On the distribution of additive and the mean values of multiplicative arithmetic functions, Studia Sci. Math. Hungar. 6 (1971), 211–233.Google Scholar
Halász, G., On the distribution of additive arithmetic functions. Collection of articles in memory of Juri Vladimirovi Linnik, Acta Arith. 27 (1975), 143–152.Google Scholar
Halberstam, H. and Richert, H.-E., Sieve methods, London Mathematical Society Monographs, vol. 4 (Academic Press, London, 1974).Google Scholar
Iwaniec, H. and Kowalski, E., Analytic number theory, American Mathematical Society Colloquium Publications, vol. 53 (American Mathematical Society, Providence, RI, 2004).Google Scholar
Korobov, N. M., Estimates of trigonometric sums and their applications, Uspekhi Mat. Nauk 13 (1958), 185–192 (in Russian).Google Scholar
Koukoulopoulos, D., On multiplicative functions which are small on average, Geom. Funct. Anal., to appear, arXiv:1111.2659v3 [math.NT].Google Scholar
Languasco, A. and Zaccagnini, A., A note on Mertens’ formula for arithmetic progressions, J. Number Theory 127 (2007), 37–46.Google Scholar
Montgomery, H. L., A note on the mean values of multiplicative functions, Report no. 17, Institut Mittag-Leffler (1978).Google Scholar
Montgomery, H. L., Ten lectures on the interface between analytic number theory and harmonic analysis, CBMS Regional Conference Series in Mathematics, vol. 84 (American Mathematical Society, Providence, RI, 1994).CrossRefGoogle Scholar
Montgomery, H. L. and Vaughan, R. C., Mean values of multiplicative functions, Period. Math. Hungar. 43 (2001), 199–214.Google Scholar
Montgomery, H. L. and Vaughan, R. C., Multiplicative number theory. I. Classical theory, Cambridge Studies in Advanced Mathematics, vol. 97 (Cambridge University Press, Cambridge, 2007).Google Scholar
Pintz, J., On Siegel’s theorem, Acta Arith. 24 (1974), 543–551.Google Scholar
Pintz, J., Elementary methods in the theory of $L$-functions. I. Hecke’s theorem, Acta Arith. 31 (1976), 53–60.Google Scholar
Selberg, A., An elementary proof of the prime-number theorem, Ann. of Math. (2) 50 (1949), 305–313.Google Scholar
Siegel, C. L., Über die Classenzahl quadratischer Zahlkörper, Acta Arith. 1 (1935), 83–86 (in German).Google Scholar
Tatuzawa, T., On a theorem of Siegel, Jpn. J. Math. 21 (1951), 163–178.Google Scholar
Tenenbaum, G., Introduction à la théorie analytique et probabiliste des nombres, Collection Échelles, third edition (Belin, Paris, 2008).Google Scholar
Titchmarsh, E. C., The theory of the Riemann zeta-function, second edition (The Clarendon Press, Oxford University Press, New York, 1986).Google Scholar
Vinogradov, I. M., A new estimate of the function $\zeta (1+ it)$, Izv. Akad. Nauk SSSR Ser. Mat. 22 (1958), 161–164 (in Russian).Google Scholar
Walfisz, A., Weylsche Exponentialsummen in der neueren Zahlentheorie, Mathematische Forschungsberichte, vol. 15 (VEB Deutscher Verlag der Wissenschaften, Berlin, 1963) (in German).Google Scholar
Wirsing, E., Elementare Beweise des Primzahlsatzes mit Restglied. II, J. Reine Angew. Math. 214/215 (1964), 1–18 (in German).Google Scholar