Title: On Arratia's coupling and the Dirichlet law for the factors of a random integer Show Chinese title

Title: 关于阿拉蒂亚耦合和随机整数因子的狄利克雷定律

Abstract: Let $x \ge 2$, let $N_x$ be an integer chosen uniformly at random from the set $\mathbb Z \cap [1, x]$, and let $(V_1, V_2, \ldots)$ be a Poisson--Dirichlet process of parameter $1$. We prove that there exists a coupling of these two random objects such that $$ \mathbb E \, \sum_{i \ge 1} |\log P_i- V_i\log x| \asymp 1, $$ where the implied constants are absolute and $N_x = P_1P_2 \cdots$ is the unique factorization of $N_x$ into primes or ones with the $P_i$'s being non-increasing. This establishes a 2002 conjecture of Arratia arXiv:1305.0941 who constructed a coupling for which the left-hand side in the above estimate is $\ll \log\!\log x$, and who also proved that the left-hand side is $\ge 1-o(1)$ for all couplings. In addition, we use our refined coupling to give a probabilistic proof of the Dirichlet law for the average distribution of the integer factorization into $k$ parts proved in 2023 by Leung arXiv:2206.14728 and we improve on its error term. Show Chinese abstract

Abstract: 设$x \ge 2$，令$N_x$从集合$\mathbb Z \cap [1, x]$中均匀随机选择一个整数，令$(V_1, V_2, \ldots)$为参数为$1$的泊松-狄利克雷过程。 我们证明存在这两个随机对象的耦合，使得$$ \mathbb E \, \sum_{i \ge 1} |\log P_i- V_i\log x| \asymp 1, $$，其中隐含的常数是绝对的，并且$N_x = P_1P_2 \cdots$是$N_x$的素数或1的唯一分解，其中$P_i$是非递增的。 这证实了 Arratia 在 2002 年提出的猜想 arXiv:1305.0941，他构造了一个耦合，使得上述估计中的左边是$\ll \log\!\log x$，他还证明了对于所有耦合，左边是$\ge 1-o(1)$。 此外，我们使用我们改进的耦合给出了整数分解为$k$部分的平均分布的 Dirichlet 定律的概率证明，该定律由 Leung 在 2023 年 arXiv:2206.14728 中证明，并我们改进了其误差项。