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How much dependence is there in the prime factorization of a random integerdistributed uniformly from 1 to n? How much dependence is there in thedecomposition into cycles of a random permutation of n points? What is therelation between the Poisson-Dirichlet process and the scale invariant Poissonprocess? These three questions have essentially the same answers, with respectto total variation distance, considering only small components, and withrespect to a Wasserstein distance, considering all components. The Wassersteindistance 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 arenecessary and sufficient to convert a uniformly distributed random integer from1 to n into a random integer prod_{p leq n} p^{Z_p} in which the multiplicityZ_p of the factor p is geometrically distributed, with all Z_p independent. Thechanges are, with probability tending to 1, one deletion, together with arandom number of insertions, having expectation 1+o(1).  The crucial tool for showing that 2+epsilon suffices is a coupling of theinfinite independent model of prime multiplicities, with the scale invariantPoisson process on (0,infty). A corollary of this construction is the firstmetric bound on the distance to the Poisson-Dirichlet in Billingsley's 1972weak 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 thei-th largest prime factor and V_i denotes the i-th component of thePoisson-Dirichlet process. It is reasonable to conjecture that O(1) isachievable.

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