Hostname: page-component-5d59c44645-jqctd Total loading time: 0 Render date: 2024-03-03T13:28:40.964Z Has data issue: false hasContentIssue false

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

Published online by Cambridge University Press:  01 March 1997

ULRICH MARTIN HIRTH
Affiliation:
Mathematics Department, Royal Holloway and Bedford New College, University of London, Egham Hill, Egham, Surrey TW20 0EX, UK

Abstract

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.

Type
Research Article
Copyright
1997 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)