On the Amount of Dependence in the Prime Factorization of a Uniform Random Integer

Abstract

How much dependence is there in the prime factorization of a random integer distributed uniformly from 1 to n? How much dependence is there in the decomposition into cycles of a random permutation of n points? What is the relation between the Poisson-Dirichlet process and the scale invariant Poisson process? These three questions have essentially the same answers, with respect to total variation distance, considering only small components, and with respect to a Wasserstein distance, considering all components. The Wasserstein distance is the expected number of changes -- insertions and deletions -- needed to change the dependent system into an independent system. In particular we show that for primes, roughly speaking, 2+o(1) changes are necessary and sufficient to convert a uniformly distributed random integer from 1 to n into a random integer prod_{p leq n} p^{Z_p} in which the multiplicity Z_p of the factor p is geometrically distributed, with all Z_p independent. The changes are, with probability tending to 1, one deletion, together with a random number of insertions, having expectation 1+o(1). The crucial tool for showing that 2+epsilon suffices is a coupling of the infinite independent model of prime multiplicities, with the scale invariant Poisson process on (0,infty). A corollary of this construction is the first metric bound on the distance to the Poisson-Dirichlet in Billingsley's 1972 weak convergence result. Our bound takes the form: there are couplings in which E sum |log P_i(n) - (log n) V_i | = O(\log \log n), where P_i denotes the i-th largest prime factor and V_i denotes the i-th component of the Poisson-Dirichlet process. It is reasonable to conjecture that O(1) is achievable.