Geodesic
Spacings Between Integers Having Typically Many Prime Factors
Canadian mathematical bulletin, Tome 53 (2010) no. 1, pp. 102-117

Voir la notice de l'article provenant de la source Cambridge University Press

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.
DOI : 10.4153/CMB-2010-022-8
Mots-clés : 11K99 
Khan, Rizwanur. Spacings Between Integers Having Typically Many Prime Factors. Canadian mathematical bulletin, Tome 53 (2010) no. 1, pp. 102-117. doi: 10.4153/CMB-2010-022-8
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[1] [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] [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:10.2307/2371483 Google Scholar

[3] [3] Gallagher, P. X., On the distribution of primes in short intervals. Mathematika 23(1976), no. 1, 4–9. Google Scholar

[4] [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.2008.04.001 Google Scholar

[5] [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] [6] Hooley, C., On the difference between consecutive numbers prime to n. II. Publ. Math. Debrecen 12(1965), 39–49. Google Scholar

[7] [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] [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/S0012-7094-99-10008-1 Google Scholar

[9] [9] Rényi, A. and Turán, P., On a theorem of Erdʺos-Kac. Acta Arith. 4(1958), 71–84. Google Scholar

[10] [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/s002220100141 Google Scholar

[11] [11] Tenenbaum, G., Introduction to analytic and probabilistic number theory. Cambridge Studies in Advanced Mathematics 46, Cambridge University Press, Cambridge, 1995. Google Scholar

[12] [12] Turán, P., On a theorem of Hardy and Ramanujan. J. London Math. Soc. (1) 9(1934), 274–276. Google Scholar

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