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

Abstract

The classical Mertens' formula states that ${\prod }_{p\le N}(1-\frac{1}{p}{)}^{-1}\sim e^{\gamma }\log N,$ 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: $$\begin{array}{rlr}& i.\underset{N\to \infty }{\lim }\frac{{\sum }_{n:p|n\Rightarrow p\le N}\frac{1}{n}}{{\sum }_{n\le N}\frac{1}{n}}=e^{\gamma }& ii.\sum _{n:p|n\Rightarrow p\le N}\frac{1}{n}\sim e^{\gamma }\log N.\end{array}$$ 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 $$\underset{N\to \infty }{\lim }\frac{{\sum }_{n:p|n\Rightarrow p\le N\text{and}p\in A}\frac{1}{n}}{{\sum }_{n\le N:p|n\Rightarrow p\in A}\frac{1}{n}}=e^{\gamma \theta }\Gamma (\theta +1),$$ and also, for any $k\ge 2$, $$\underset{N\to \infty }{\lim }\frac{{\sum }_{n:p|n\Rightarrow p\le N\text{and}p\in A}^{{}^{\prime }(k)}\frac{1}{n}}{{\sum }_{n\le N:p|n\Rightarrow p\in A}^{{}^{\prime }(k)}\frac{1}{n}}=e^{\gamma \theta }\Gamma (\theta +1),$$ where ${\sum }^{{}^{\prime }(k)}$ denotes that the summation is restricted to $k$-free positive integers. In the spirit of (ii), we show for example that ${\sum }_{n:p|n\Rightarrow p\le N}^{{}^{\prime }(k)}\frac{1}{n_{\{(k-1)-\text{free}\}}\varphi (n_{\{(k-1)-\text{power}\}})}\sim e^{\gamma }\log N,$ where $\varphi$ 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 $n$.

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