COUPLINGS, COMPONENT COUNTING PROCESSES, AND PROBABILISTIC NUMBER THEORY

Our results are concerned with couplings, component counts of combinatorial objects, andprobabilistic number theory. In the theory of couplings, we are concerned with the generalproblem of proving the existence of joint distributions p(i,j) of two discrete random variables M andN subject to infinitely many constraints of the form p(M=i, N=j)=0. The constraints placed on the joint distributions will require, for manyelements j in the range of N, p(M=i, N=j)=0 for infinitely many values of i in therange of M, where the corresponding values of i depend on j. To prove the existence ofsuch joint distributions, we apply a theorem proved by Volker Strassen on the existence of jointdistributions with prespecified marginal distributions. In the case in which N is uniformlydistributed in a combinatorial structure with C_i components of size i, we seek to measurethe amount of dependence in the process (C_i )_{i\le n} by coupling N with a variable M such thatM has Z_i components of size i and the Z_i's are independent, withsum_{i\le n}(C_i - Z_i)^+ <2.In the combinatorial example of noncrossing partitions, we providetwo derivations of the probability distribution of the component counts of a uniformly distributed noncrossing partition. Upon applying a bijection between the set of noncrossing partitionsand Dyck paths consisting of up-steps and down-steps, our results specify the joint and marginal distributions of the block counts of the number of consecutive up-steps in a uniformly random chosen Dyck path.In number theory, we give an analogue of the Erd\"{o}s-KacTheorem by providing a family of integer-valued random variables on{1,\ldots,n} whose number of distinct prime factorsis roughly log\log n+X\cdot\sqrt{\log\log n} for large valuesof n, where X is a standard normal variable. Our final resultinvolves couplings of a Zeta-distributed variable. Given s>1 andn\in\mathbb{N}, consider a Zeta(s)-distributed integer-valuedrandom variable with prime factorization Z(s)=\prod_{p}p^{\alpha_{p}(s)}and the truncation Z_{n}(s)\coloneqq\prod_{p\le n}p^{\alpha_{p}(s)The prime powers \alpha_{p}(s) are independent with\alpha_{p}(s)\sim\text{Geometric}(1/p^s),and we also consider a random variable M(n)=\prod_{p\le n}p^{Z_{p}},where the Z_{p}'s are independent with Z_{p}\sim\text{Geometric}(1/p).We apply the concept of pivot mass and a theorem proved by Strassenin order to prove the existence of couplings of a Zeta-distributed random variable Z(s) and M(n)in which we can make probabilistic divisibility statements of the form "Z_{n}(s) divides M(n)P(n)"for some random prime P(n)\le n. In particular, we willprove that for each n\in\mathbb{N} and an integer k\ge 4, there exists an \varepsilon(k)>0such that when s\in\left(1,1+\varepsilon(k) we cancouple Z(s) and M\left(n\right) such that if Z(s)\le k,then Z_{n}(s) always divides M(n)P(n)for some random prime P(n)\le n