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Probabilistic Number Theory, the GEM/Poisson-Dirichlet Distribution and the Arc-sine Law

Published online by Cambridge University Press:  01 March 1997

Mathematics Department, Royal Holloway and Bedford New College, University of London, Egham Hill, Egham, Surrey TW20 0EX, UK


The prime factorization of a random integer has a GEM/Poisson-Dirichlet distribution as transparently proved by Donnelly and Grimmett [8]. By similarity to the arc-sine law for the mean distribution of the divisors of a random integer, due to Deshouillers, Dress and Tenenbaum [6] (see also Tenenbaum [24, II.6.2, p. 233]), – the ‘DDT theorem’ – we obtain an arc-sine law in the GEM/Poisson-Dirichlet context. In this context we also investigate the distribution of the number of components larger than ε which correspond to the number of prime factors larger than nε.

Research Article
1997 Cambridge University Press

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