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February 2011 Another proof on the existence of Mertens’s constant
Marek Wójtowicz
Proc. Japan Acad. Ser. A Math. Sci. 87(2): 22-23 (February 2011). DOI: 10.3792/pjaa.87.22

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.

Citation

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Marek Wójtowicz. "Another proof on the existence of Mertens’s constant." Proc. Japan Acad. Ser. A Math. Sci. 87 (2) 22 - 23, February 2011. https://doi.org/10.3792/pjaa.87.22

Information

Published: February 2011
First available in Project Euclid: 1 February 2011

zbMATH: 1252.11006
MathSciNet: MR2797580
Digital Object Identifier: 10.3792/pjaa.87.22

Subjects:
Primary: 11A41

Keywords: Mertens’s constant , prime number

Rights: Copyright © 2011 The Japan Academy

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Vol.87 • No. 2 • February 2011
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