Hostname: page-component-745bb68f8f-f46jp Total loading time: 0 Render date: 2025-01-13T11:12:52.146Z Has data issue: false hasContentIssue false
  • English
  • Français

Number of prime factors with a given multiplicity

Part of: Elementary number theory

Published online by Cambridge University Press:  03 May 2021

Ertan Elma  and
Yu-Ru Liu
Ertan Elma*
Affiliation:
Department of Pure Mathematics, University of Waterloo, 200 University Avenue West, Waterloo, ONN2L 3G1, Canadayrliu@uwaterloo.ca
Yu-Ru Liu
Affiliation:
Department of Pure Mathematics, University of Waterloo, 200 University Avenue West, Waterloo, ONN2L 3G1, Canadayrliu@uwaterloo.ca
*
e-mail: eelma@uwaterloo.ca
Get access Rights & Permissions [Opens in a new window]

Abstract

Let $k\geqslant 1$ be a natural number and $\omega _k(n)$ denote the number of distinct prime factors of a natural number n with multiplicity k. We estimate the first and second moments of the functions $\omega _k$ with $k\geqslant 1$ . Moreover, we prove that the function $\omega _1(n)$ has normal order $\log \log n$ and the function $(\omega _1(n)-\log \log n)/\sqrt {\log \log n}$ has a normal distribution. Finally, we prove that the functions $\omega _k(n)$ with $k\geqslant 2$ do not have normal order $F(n)$ for any nondecreasing nonnegative function F.

Keywords

Prime divisorsnormal ordernormal distribution

MSC classification

Primary: 11A25: Arithmetic functions; related numbers; inversion formulas
Secondary: 11A41: Primes
Type
Article
Information
Canadian Mathematical Bulletin , Volume 65 , Issue 1 , March 2022 , pp. 253 - 269
Check for updates
Copyright
© Canadian Mathematical Society 2021

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.)

References

Billingsley, P., On the central limit theorem for the prime divisor functions. Amer. Math. Monthly. 76(1969), 132–139.10.1080/00029890.1969.12000157CrossRefGoogle Scholar
Delange, H., Sur le nombre des diviseurs premiers de $n$ . C. R. Acad. Sci. Paris 237(1953), 542–544.Google Scholar
Delange, H., Sur un théorème d’Erdős et Kac. Acad. Roy. Belg. Bull. Cl. Sci. (5) 42(1956), 130–144.Google Scholar
Delange, H., Sur les fonctions arithmétiques fortement additives. C. R. Acad. Sci. Paris 244(1957), 2122–2124.Google Scholar
Delange, H., Sur des formules de Atle Selberg. Acta Arith. 19(1971), 105–146.10.4064/aa-19-2-105-146CrossRefGoogle Scholar
Elliott, P. D. T. A., Probabilistic number theory I: mean value theorem. Grundlehren der mathematischen Wissenschaften, 239, Springer, New York, 1979.10.1007/978-1-4612-9989-9CrossRefGoogle Scholar
Elliott, P. D. T. A., Probabilistic number theory II: central limit theorems. Grundlehren der mathematischen Wissenschaften, 240, Springer, New York, 1979.10.1007/978-1-4612-9989-9CrossRefGoogle Scholar
Erdős, P. and Kac, M., The Gaussian law of errors in the theory of additive number theoretic functions. Amer. J. Math. 62(1940), 738–742.10.2307/2371483CrossRefGoogle Scholar
Granville, A. and Soundararajan, K., Sieve and the Erdős—Kac theorem, equidistribution in number theory: an introduction. NATO Sci. Ser. II Math. Phys. Chem. 237(2007), 15–27.Google Scholar
Hardy, G. H. and Ramanujan, S., The normal number of prime factors of a number n. Quart. J. Math. 48(1917), 76–92. In: Collected papers of Srinivasa Ramanujan, AMS Chelsea Publishing, Providence, RI, 2000, pp. 562–575.Google Scholar
Hardy, G. H. and Wright, E. M., An introduction to the theory of numbers. 6th ed., Oxford University Press, Oxford, UK, 2008, revised by D. R. Heath-Brown and J. H. Silverman, with a foreword by Andrew Wiles.Google Scholar
Ivić, A., The Riemann zeta-function. Dover Publications, Inc., Mineola, NY, 2003, theory and applications, reprint of the 1985 original (Wiley, New York).Google Scholar
Kubilius, J., Probabilistic methods in number theory. Translations of Mathematical Monographs, 11, American Mathematical Society, Providence, RI, 1964.10.1090/mmono/011CrossRefGoogle Scholar
Misevičius, G., The use of the method of moments in probabilistic number theory (Russian, Lithuanian). Litovsk. Mat. Sb. 5(1965), 275–289.Google Scholar
Montgomery, H. L. and Vaughan, R. C., Multiplicative number theory I: classical theory. Cambridge Studies in Advanced Mathematics, 97, Cambridge University Press, Cambridge, MA, 2007.Google Scholar
Murty, M. R., Problems in analytic number theory. 2nd ed., Graduate Texts in Mathematics, 206, Springer, New York, 2008, readings in Mathematics.Google Scholar
Saidak, F., An elementary proof of a theorem of Delange. C. R. Math. Acad. Sci. Soc. R. Can. 24(2002), no. 4, 144–151.Google Scholar
Sathe, L. G., On a problem of Hardy on the distribution of integers having a given number of prime factors (I.–IV.). J. Indian Math. Soc. (N.S.) 17(1953), 63–82, 83–141, and 18(1954), 27–42, 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
Shapiro, H., Distribution functions of additive arithmetic functions. Proc. Nat. Acad. Sci. 42(1956), 426–430.10.1073/pnas.42.7.426CrossRefGoogle ScholarPubMed
Titchmarsh, E. C., The theory of the Riemann zeta-function. 2nd ed., The Clarendon Press and Oxford University Press, New York, 1986, edited and with a preface by D. R. Heath-Brown.Google Scholar
Turán, P., On a theorem of Hardy and Ramanujan. J. Lond. Math. Soc. 9(1934), no. 4, 274–276.10.1112/jlms/s1-9.4.274CrossRefGoogle Scholar
Vilkas, E., On the normal distribution of additive arithmetical functions (in Russian). Uc. Trudy viln. Gosud. Inst. 16(1957), 23–31.Google Scholar