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A Generalization of the Erdös-Kac Theorem and its Applications

Published online by Cambridge University Press:  20 November 2018

Yu-Ru Liu
Yu-Ru Liu*
Affiliation:
Department of Pure Mathematics University of Waterloo Waterloo, ON N2L 3G1, e-mail: yrliu@math.uwaterloo.ca
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Abstract

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We axiomatize the main properties of the classical Erdös-Kac Theorem in order to apply it to a general context. We provide applications in the cases of number fields, function fields, and geometrically irreducible varieties over a finite field.

Keywords

11N6011N80
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 47 , Issue 4 , 01 December 2004 , pp. 589 - 606
Copyright
Copyright © Canadian Mathematical Society 2004

References

[1] Billingsley, P., On the central limit theorem for the prime divisor functions. Amer.Math. Monthly 76 (1969), 132–139.Google Scholar
[2] 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.Google Scholar
[3] Feller, W., An introduction to probability theory and its applications, Vol. II. Wiley, New York, 1966.Google Scholar
[4] Halberstam, H., On the distribution of additive number theoretic functions, I–III. J. LondonMath. Soc. 30 (1955), 43–53; 31 (1956), 1–14, 14–27.Google Scholar
[5] Hardy, G. H. and Ramanujan, S., The normal number of prime factors of a number n. Quar. J. Pure Appl. Math. 48 (1917), 76–97.Google Scholar
[6] Knopfmacher, J., Analytic arithmetic of algebraic function fields. Lecture Notes in Pure and Applied Math. 50(1979).Google Scholar
[7] Landau, E., Neuer Beweis des Primzahlsatzes und Beweis des Primidealsatzes. Math. Ann. 56 (1903), 645–670.Google Scholar
[8] Lang, S. and Weil, A., Number of points of varieties in finite fields. Amer. J. Math. 76 (1954), 819–827.Google Scholar
[9] Liu, Y.-R., A generalization of the Turán Theorem and its applications. Canad. Math. Bulletin. 47 (2004), 573–588.Google Scholar
[10] Lorenzini, D., An invitation to arithmetic geometry. Graduate Studies in Mathematics 9, American Mathematical Society, Providence, RI, 1996.Google Scholar
[11] Murty, M. R., Problems in analytic number theory. Graduate Texts in Mathematics 206, Springer-Verlag, New York, 2001.Google Scholar
[12] Turán, P., On a theorem of Hardy and Ramanujan. J. LondonMath. Soc. 9 (1934), 274–276.Google Scholar
[13] Weber, H., Über Zahlengruppen in algebraischen Körpern. Math. Ann. 49 (1897), 83–100.Google Scholar
[14] Zhang, W.-B., probabilistic number theory in additive arithmetic semigroup I. Analytic Number Theory, Vol 2, Progress in Mathematics 139, Birkhauser, Boston, MA, 1995, pp. 839–885.Google Scholar