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

arXiv1809.04888MaRDI QIDQ6306676

Ross G. Pinsky

Publication date: 13 September 2018

Abstract: The classical Mertens' formula states that $$ 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: ${\displaystyle }$ �egin{aligned} &i. lim_{N oinfty}frac{sum_{n:p|nRightarrow ple N} hinspacefrac1n}{sum_{nle N}frac1n}=e^gamma &ii. sum_{n:p|nRightarrow ple N} hinspacefrac1nsim e^gammalog N. end{aligned} ${\displaystyle }$ 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 $hetain(0,1]$ with respect to the set of all primes, then ${\displaystyle }$ lim_{N oinfty}frac{sum_{n:p|nRightarrow ple N hinspace ext{and} hinspace pin A}frac1n} {sum_{nle N:p|nRightarrow pin A}frac1n}=e^{gamma heta}Gamma( heta+1), ${\displaystyle }$ and also, for any $kge2$, ${\displaystyle }$ lim_{N oinfty}frac{sum^{'(k)}_{n:p|nRightarrow ple N hinspace ext{and} hinspace pin A}frac1n} {sum^{'(k)}_{nle N:p|nRightarrow pin A}frac1n}=e^{gamma heta}Gamma( heta+1), ${\displaystyle }$ where $su{m}^{\text{'}(k)}$ denotes that the summation is restricted to $k$-free positive integers. In the spirit of (ii), we show for example that $su{m}_{n:p|nRightarrowpleN}^{\text{'}(k)}frac1n_{(k-1)-extfree}phi(n_{(k-1)-extpower})sim{e}^{g}ammalogN,$ where $phi$ is the Euler totient function, and $n_{(k-1)-extfree}$ and $n_{(k-1)-extpower}$ are the $(k-1)$-free part and the $(k-1)$-power part of $n$.

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