Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-23T07:06:51.866Z Has data issue: false hasContentIssue false
  • English
  • Français

Spacings Between Integers Having Typically Many Prime Factors

Published online by Cambridge University Press:  20 November 2018

Rizwanur Khan
Rizwanur Khan*
Affiliation:
University of Michigan, Department of Mathematics, Ann Arbor, MI 48109, USA e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We show that the sequence of integers which have nearly the typical number of distinct prime factors forms a Poisson process. More precisely, for $\delta$ arbitrarily small and positive, the nearest neighbor spacings between integers $n$ with $\left| \omega \left( n \right)\,-\,\log \log n \right|\,<\,{{\left( \log \log n \right)}^{\delta }}$ obey the Poisson distribution law.

Keywords

11K99
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 53 , Issue 1 , 01 March 2010 , pp. 102 - 117
Copyright
Copyright © Canadian Mathematical Society 2010

References

[1] Cobeli, C. and Zaharescu, A., On the distribution of primitive roots mod p. Acta Arith. 83(1998), no. 2, 143–153.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), pp. 738–742. doi:10.2307/2371483Google Scholar
[3] Gallagher, P. X., On the distribution of primes in short intervals. Mathematika 23(1976), no. 1, 4–9.Google Scholar
[4] Granville, A. and Kurlberg, P., Poisson statistics via the Chinese Remainder Theorem. Adv. Math. 218(2008), no. 6, 2013–2042. doi:10.1016/j.aim.2008.04.001Google Scholar
[5] Granville, A. and Soundararajan, K., Sieving and the Erdös-Kac theorem. In: Equidistribution in number theory, an introduction, NATO Sci. Ser. II Math. Phys. Chem. 237, Springer, Dordrecht, 2007, pp. 15–27.Google Scholar
[6] Hooley, C., On the difference between consecutive numbers prime to n. II. Publ. Math. Debrecen 12(1965), 39–49.Google Scholar
[7] Khan, R., On the distribution of ω(n) . In: Anatomy of Integers. CRM Proc Lecture Notes 46, American Mathematical Society, Providence, RI, 2008, pp. 199–207.Google Scholar
[8] Kurlberg, P. and Rudnick, Z., The distribution of spacings between quadratic residues. Duke Math. J. 100(1999), no. 2, 211–242. doi:10.1215/S0012-7094-99-10008-1Google Scholar
[9] Rényi, A. and Turán, P., On a theorem of Erdʺos-Kac. Acta Arith. 4(1958), 71–84.Google Scholar
[10] Rudnick, Z., Sarnak, P., and Zaharescu, A., The distribution of spacings between the fractional parts of n 2α . Invent. Math. 145(2001), no. 1, 37–57. doi:10.1007/s002220100141Google Scholar
[11] Tenenbaum, G., Introduction to analytic and probabilistic number theory. Cambridge Studies in Advanced Mathematics 46, Cambridge University Press, Cambridge, 1995.Google Scholar
[12] Turán, P., On a theorem of Hardy and Ramanujan. J. London Math. Soc. (1) 9(1934), 274–276.Google Scholar