Generalization of Erdős-Kac theorem

Abstract

Let ω(n) is the number of distinct prime factors of the natural number n,we consider two cases where \ell 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

References

[1] Delange H. Sur les fonctions arithmétiques multiplicatives. Ann Sci Éc Norm Supér, 1961, 78: 273–304

[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

[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

[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

[5] Halberstam H. On the distribution of additive number theoretic functions (I). J Lond Math Soc, 1955, 30: 43–53

[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

[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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