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On the Largest Prime Divisors of an Integer

In: Extreme Value Theory and Applications

Author

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  • Jean-Marie De Koninck

    (Université Laval, Départment de mathématiques et de statistique)

Abstract

Several results of number theory can be expressed in probabilistic terms and, for others, the simplest proof is by probabilistic methods. Simply take the uniform distribution on the consecutive integers 1, 2,…, N. Then arithmetic functions, when restricted to the integers 1 through N, become random variables and arithmetic means are expectations. The power of probabilistic methods lies in the fact that divisibility by distinct primes are almost independent events. On the other hand, most problems remain challenging since the errors generated by the not exact independence can be dominating in a problem when one faces an increasing number of primes. The best example is the study of large prime divisors where the results do not resemble those which one would get for independent random variables.

Suggested Citation

  • Jean-Marie De Koninck, 1994. "On the Largest Prime Divisors of an Integer," Springer Books, in: Janos Galambos & James Lechner & Emil Simiu (ed.), Extreme Value Theory and Applications, pages 447-462, Springer.
    • Handle: RePEc:spr:sprchp:978-1-4613-3638-9_27
    DOI: 10.1007/978-1-4613-3638-9_27
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