Generalization of Erdős-Kac theorem

Yalin SUN, Lizhen WU

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PDF(283 KB)
Front. Math. China ›› 2019, Vol. 14 ›› Issue (6) : 1303-1316. DOI: 10.1007/s11464-019-0808-2
RESEARCH ARTICLE
RESEARCH ARTICLE

Generalization of Erdős-Kac theorem

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Abstract

Let ω(n) is the number of distinct prime factors of the natural number n,we consider two cases where is even and odd natural numbers, and then we prove a more general form of the classical Erdős-Kac theorem.

Keywords

Erdős-Kac theorem / central limit theorem / value distribution

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Yalin SUN, Lizhen WU. Generalization of Erdős-Kac theorem. Front. Math. China, 2019, 14(6): 1303‒1316 https://doi.org/10.1007/s11464-019-0808-2

References

[1]
Delange H. Sur les fonctions arithmétiques multiplicatives. Ann Sci Éc Norm Supér, 1961, 78: 273–304
CrossRef Google scholar
[2]
Erdős P, Kac M. On the Gaussian law of error in the theory of additive functions. Proc Natl Acad Sci USA, 1939, 25: 206–207
CrossRef Google scholar
[3]
Erdős P, Kac M. The Gaussian law of errors in the theory of additive number theoretic functions. Amer J Math, 1940, 62: 738–742
CrossRef Google scholar
[4]
Granville A, Soundararajan K. Sieving and the Erdős-Kac theorem. In: Granville A, Rudnick Z, eds. Equidistribution in Number Theory, An Introduction. NATO Science Series, Sub-Series II: Mathematics, Physics and Chemistry, Vol 237. Dordrecht: Springer, 2007, 15–27
CrossRef Google scholar
[5]
Halberstam H. On the distribution of additive number theoretic functions (I). J Lond Math Soc, 1955, 30: 43–53
CrossRef Google scholar
[6]
Hardy G H, Ramanujan S. The normal number of prime factors of a number n.Quart J Math, 1917, 48: 76–92
[7]
Hardy G H, Wright E M. An Introduction to the Theory of Numbers. Oxford: Oxford Univ Press, 1979
[8]
Kac M. Probability methods in some problems of analysis and number theory. Bull Amer Math Soc, 1949, 55: 641–665
CrossRef Google scholar
[9]
Sathe L G. On a problem of Hardy and Ramanujan on the distribution of integers having a given number of prime factors. J Indian Math Soc, 1953, 17: 63–141
[10]
Sathe L G. On a problem of Hardy and Ramanujan on the distribution of integers having a given number of prime factors. J Indian Math Soc, 1954, 18: 27–81
[11]
Selberg A. Note on the paper by L. G. Sathe. J Indian Math Soc, 1954, 18: 83–87
[12]
Tenenbaum G. Introduction to Analytic and Probabilistic Number Theory. Cambridge Stud Adv Math, Vol 46. Cambridge: Cambridge Univ Press, 1995

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