标题： 解析数论中Selberg--Delange方法的概率解释 显示英文标题

标题： Probabilistic interpretation of the Selberg--Delange Method in analytic number theory

摘要： 在解析数论中，Selberg--Delange 方法为一个复函数$f$的部分和提供了渐近公式，其狄利克雷级数的形式是具有良好行为的解析函数与黎曼 zeta 函数的复次幂的乘积。 在概率论中，模泊松收敛是一种向正态分布收敛的改进形式。 这种更强形式的收敛不仅意味着中心极限定理，还对变量的分布提供了更精细的控制，例如对大偏差的精确估计。 在本文中，我们表明使用 Selberg--Delange 方法得到的解析数论结果导致了模泊松收敛，当$x \to \infty$时，对于在$1$和$x$之间随机选择的整数的不同素因子个数，其中整数按照一类广泛的乘法函数分布。 作为推论，我们在不同的但相关的条件下重新获得了 Elboim 和 Gorodetsky 最近结果的一部分：此类随机整数的不同素因子个数的中心极限定理。 显示英文摘要

摘要： In analytic number theory, the Selberg--Delange Method provides an asymptotic formula for the partial sums of a complex function $f$ whose Dirichlet series has the form of a product of a well-behaved analytic function and a complex power of the Riemann zeta function. In probability theory, mod-Poisson convergence is a refinement of convergence in distribution toward a normal distribution. This stronger form of convergence not only implies a Central Limit Theorem but also offers finer control over the distribution of the variables, such as precise estimates for large deviations. In this paper, we show that results in analytic number theory derived using the Selberg--Delange Method lead to mod-Poisson convergence as $x \to \infty$ for the number of distinct prime factors of a randomly chosen integer between $1$ and $x$, where the integer is distributed according to a broad class of multiplicative functions. As a Corollary, we recover a part of a recent result by Elboim and Gorodetsky under different, though related, conditions: A Central Limit Theorem for the number of distinct prime factors of such random integers.