Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-13T05:54:32.721Z Has data issue: false hasContentIssue false
  • English
  • Français

Joint Poisson distribution of prime factors in sets

Part of: Multiplicative number theory

Published online by Cambridge University Press:  23 June 2021

KEVIN FORD
KEVIN FORD*
Affiliation:
Department of Mathematics, 1409 West Green Street, University of Illinois at Urbana-Champaign, Urbana, IL 61801, U.S.A. e-mail: ford@math.uiuc.edu
Get access Rights & Permissions [Opens in a new window]

Abstarct

Given disjoint subsets T1, …, Tm of “not too large” primes up to x, we establish that for a random integer n drawn from [1, x], the m-dimensional vector enumerating the number of prime factors of n from T1, …, Tm converges to a vector of m independent Poisson random variables. We give a specific rate of convergence using the Kubilius model of prime factors. We also show a universal upper bound of Poisson type when T1, …, Tm are unrestricted, and apply this to the distribution of the number of prime factors from a set T conditional on n having k total prime factors.

MSC classification

Primary: 11N25: Distribution of integers with specified multiplicative constraints
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 173 , Issue 1 , July 2022 , pp. 189 - 200
Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Cambridge Philosophical Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

†

Supported by NSF grant DMS-1802139.

References

Ash, R. B.. Information Theory. Corrected reprint of the 1965 original. (Dover Publications, Inc., New York, 1990). xii+339 pp.Google Scholar
Barbour, A. D., Holst, L. and Janson, S.. Poisson approximation. Oxford Studies in Probability, 2. Oxford Science Publications (The Clarendon Press, Oxford University Press, New York, 1992).Google Scholar
Dartyge, C. and Tenenbaum, G.. Sommes des chiffres de multiples d’entiers. (French. English, French summary) [Sums of digits of multiples of integers] Ann. Inst. Fourier (Grenoble) 55 (2005), no. 7, 2423–2474.10.5802/aif.2166CrossRefGoogle Scholar
Delange, H.. Sur des formules de Atle Selberg, Acta Arith. 19 (1971), 105–146.CrossRefGoogle Scholar
Elliott, P. D. T. A.. Probabilistic number theory. II. Central limit theorems. Grundlehren Math. Wiss. [Fundamental Principles of Mathematical Sciences], 240 (Springer–Verlag, Berlin-New York, 1980).CrossRefGoogle Scholar
Erdös, P.. Note on the number of prime divisors of integers. J. London Math. Soc. 12 (1937), 308–314.10.1112/jlms/s1-12.48.308CrossRefGoogle Scholar
Erdös, P. and Kac, M.. The Gaussian law of errors in the theory of additive number theoretic functions. Amer. J. Math. 62 (1940), 738–742.CrossRefGoogle 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
Hall, R. R. and Tenenbaum, G.. Divisors, Cambridge Tracts in Mathematics (Cambridge University Press, Cambridge, 1988 Vol 90).Google Scholar
Hardy, G. H. and Ramanujan, S.. The normal number of prime factors of a number n, Quart. J. Math. Oxford 48 (1917), 76–92.Google Scholar
Kubilius, J.. Probabilistic methods in the theory of numbers. Uspehi Mat. Nauk (N.S.) 11 (1956), 2(68), 31–66 (Russian); = Amer. Math. Soc. Translations, 19 (1962), 47–85.Google Scholar
Kubilius, J.. Probabilistic methods in the theory of numbers Transl. Math. Monogr. vol. 11 (American Mathematical Society, Providence, R.I. 1964).CrossRefGoogle Scholar
Landau, E.. Handbuch der Lehre von der Verteilung der Primzahlen (Chelsea, 1951). Reprint of the 1909 original.Google Scholar
Mangerel, A. P.. Topics in multiplicative and probabilistic number theory, PhD. thesis University of Toronto (2018).Google Scholar
Sárközy, A.. Remarks on a paper of G. Halász: “On the distribution of additive and the mean values of multiplicative arithmetic functions” (Studia Sci. Math. Hungar. 6 (1971), 211–233). Period. Math. Hungar. 8 (1977), no. 2, 135–150.Google Scholar
Sathe, L. G.. On a problem of Hardy on the distribution of integers having a given number of prime factors. II. J. Indian Math. Soc. (N.S.) 17 (1953), 83–141; III. ibid, 18 (1954), 27–42; IV. ibid, 18 (1954), 43–81.Google Scholar
Selberg, A.. Note on a paper by L. G. Sathe. J. Indian Math. Soc. (N.S.) 18 (1954), 83–87.Google Scholar
Tenenbaum, G.. Crible d’Ératosthène et modèle de Kubilius. (French. English summary) [The sieve of Eratosthenes and the model of Kubilius] Number Theory in Progress, vol. 2 (Zakopane–Kościelisko, 1997), 1099–1129 (de Gruyter, Berlin, 1999).CrossRefGoogle Scholar
Tenenbaum, G.. A rate estimate in Billingsley’s theorem for the size distribution of large prime factors. Quart. J. Math. Oxford 51 (2000), no. 3, 385–403.Google Scholar
Tenenbaum, G.. Introduction to analytic and probabilistic number theory. Graduate Studies in Mathematics, vol. 163 (American Mathematical Society, Providence, RI, third edition, 2015). Translated from the 2008 French edition by Patrick D. F. Ion.CrossRefGoogle Scholar
Tenenbaum, G.. Moyennes effectives de fonctions multiplicatives complexes. (French. English summary) [Effective means for complex multiplicative functions] Ramanujan J. 44 (2017), no. 3, 641–701. Errata: to appear in the Ramanujan J., also available on the author’s web page: http://www.iecl.univ-lorraine.fr/ Gerald.Tenenbaum/PUBLIC/Prepublications_et_publications/CrossRefGoogle Scholar
Tudesq, C.. Majoration de la loi locale de certaines fonctions additives. Arch. Math. (Basel), 67(6), (1996), 465–472.CrossRefGoogle Scholar