On the Largest Prime Divisors of an Integer

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.