Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

The intermediate prime divisors of integers


Authors: J.-M. De Koninck and J. Galambos
Journal: Proc. Amer. Math. Soc. 101 (1987), 213-216
MSC: Primary 11K99
DOI: https://doi.org/10.1090/S0002-9939-1987-0902529-X
MathSciNet review: 902529
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ {p_1} < {p_2} < \cdots < {p_\omega }$ be the distinct prime divisors of the integer $ n$. If $ \omega = \omega (n) \to + \infty $ with $ n$, then $ {p_j}$ is called an intermediate prime divisor of $ n$ if both $ j$ and $ \omega - j$ tend to infinity with $ n$. We show that $ \log \log {p_j}$, as $ j$ goes through the indices for which $ {p_j}$ is intermediate, forms a limiting Poisson process in the sense of natural density.


References [Enhancements On Off] (What's this?)

  • [1] J.-M. De Koninck and A. Ivic, Topics in arithmetical functions, Notas de Matematica 72, North-Holland, Amsterdam, 1980. MR 589545 (82a:10047)
  • [2] P. D. T. A. Elliott, Probabilistic number theory. I, Springer-Verlag, Berlin and New York, 1979. MR 551361 (82h:10002a)
  • [3] -, Probabilistic number theory. II, Springer-Verlag, Berlin and New York, 1980. MR 560507 (82h:10002b)
  • [4] J. Galambos, The sequences of prime divisors of integers, Acta Arith. 31 (1976), 213-218. MR 0439795 (55:12677)
  • [5] -, Introductory probability theory, Marcel Dekker, New York, 1984. MR 772381 (86i:60002)
  • [6] G. H. Hardy and E. M. Wright, An introduction to the theory of numbers, 3rd ed., Oxford Univ. Press, 1960. MR 0067125 (16:673c)
  • [7] H. Maier, On the set of divisors of an integer, Technical Report, Univ. of Michigan, Ann Arbor, 1983.
  • [8] A. Rényi, Remarks on the Poisson process, Studia Sci. Math. Hungar. 2 (1967), 119-123. MR 0212861 (35:3726)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 11K99

Retrieve articles in all journals with MSC: 11K99


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1987-0902529-X
Article copyright: © Copyright 1987 American Mathematical Society

American Mathematical Society