Probabilistic Proofs of Some Generalized Mertens' Formulas Via Generalized Dickman Distributions

TL;DR

This paper introduces probabilistic methods and generalized Dickman distributions to derive new and existing generalized Mertens' formulas, expanding understanding of prime-related products and sums in number theory.

Contribution

It develops probabilistic proofs for generalized Mertens' formulas using weak convergence to generalized Dickman distributions, providing new insights and extensions beyond traditional number-theoretic approaches.

Findings

Derived limits involving subsets of primes with natural density.

Established asymptotic formulas for sums over k-free integers.

Connected probabilistic models with classical number-theoretic results.

Abstract

The classical Mertens' formula states that $ \prod_{p\le N}\big(1-\frac1p)^{-1}\sim e^\gamma\log N, $ where the product is over all primes $p$ less than or equal to $N$, and $\gamma$ is the Euler-Mascheroni constant. By the Euler product formula, this is equivalent to either of the following statements: $$ \begin{aligned} &i. \lim_{N\to\infty}\frac{\sum_{n:p|n\Rightarrow p\le N}\thinspace\frac1n}{\sum_{n\le N}\frac1n}=e^\gamma\ \ &ii. \sum_{n:p|n\Rightarrow p\le N}\thinspace\frac1n\sim e^\gamma\log N. \end{aligned} $$ Via some random integer constructions and a criterion for weak convergence of distributions to so-called generalized Dickman distributions, we obtain some generalized Mertens' formulas, some of which are new and some of which have been proved using number-theoretic tools. For example, in the spirit of (i), we show that if $A$ is a subset of the primes which has natural density $\theta\in(0,1]$ with respect to the set of all primes, then $$ \lim_{N\to\infty}\frac{\sum_{n:p|n\Rightarrow p\le N\thinspace\text{and}\thinspace p\in A}\frac1n} {\sum_{n\le N:p|n\Rightarrow p\in A}\frac1n}=e^{\gamma\theta}\Gamma(\theta+1), $$ and also, for any $k\ge2$, $$ \lim_{N\to\infty}\frac{\sum^{'(k)}_{n:p|n\Rightarrow p\le N\thinspace\text{and}\thinspace p\in A}\frac1n} {\sum^{'(k)}_{n\le N:p|n\Rightarrow p\in A}\frac1n}=e^{\gamma\theta}\Gamma(\theta+1), $$ where $\sum^{'(k)}$ denotes that the summation is restricted to $k$-free positive integers. In the spirit of (ii), we show for example that $ \sum^{'(k)}_{n:p|n\Rightarrow p\le N}\frac1{n_{\{(k-1)-\text{free}\}}\phi(n_{\{(k-1)-\text{power}\}})}\sim e^\gamma\log N, $ where $\phi$ is the Euler totient function, and $n_{\{(k-1)-\text{free}\}}$ and $n_{\{(k-1)-\text{power}\}}$ are the $(k-1)$-free part and the $(k-1)$-power part of $n$.