Another proof on the existence of Mertens’s constant

Abstract

Let $\mathcal{P}$ denote the set of all prime numbers, and let $p_{k}$ denothe the $k$th prime. In 1873 Mertens presented a quantitative proof of the divergence of the series $\sum_{p\in \mathcal{P}}\frac{1}{p}$ by showing the limit $B:= \lim_{x\to\infty}(\sum_{p\le x}\frac{1}{p} - \log\log x)$ exists with $B=0.26149\ldots$. In this paper we give another proof on the divergence of the above series. We prove the following

Theorem The sequence $f(n):= \sum_{k=1}^{n}\frac{1}{p_{k}} - \log\log n$, $n=2,3, \ldots$, is decreasing and bounded from below, and its limit equals the Mertens’s constant $B$.

In proofs of the first two conditions we use only classical estimations for $p_{k}$, obtained in 1939 and 1962 by Rosser and Schoenfeld.