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On Mertens' Theorem for Beurling Primes

Published online by Cambridge University Press:  20 November 2018

Paul Pollack
Paul Pollack*
Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, BC V6T 1Z2 e-mail: pollack@math.ubc.ca
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Abstract.

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Let $1\,<\,{{p}_{1}}\,\le \,{{p}_{2}}\,\le \,{{p}_{3}}\,\le \,...$ be an infinite sequence $P$ of real numbers for which ${{p}_{i}}\,\to \,\infty $, and associate with this sequence the Beurling zeta function$\zeta P\left( s \right)\,:=\,{{\prod\nolimits_{i=1}^{\infty }{\left( 1\,-\,p_{i}^{-s} \right)}}^{-1}}$. Suppose that for some constant $A\,>\,0$, we have $\zeta P\left( s \right)\tilde{\ }A/\left( s-1 \right),\ \text{as}\,s\,\downarrow \,1$. We prove that $P$ satisfies an analogue of a classical theorem of Mertens: ${{\prod{_{{{p}_{i}}\le x}\left( 1\,-\,{1}/{{{p}_{i}}}\; \right)}}^{-1}}\,\sim \,A{{\text{e}}^{\gamma }}\,\log \,x$, as $x\,\to \,\infty $. Here $\text{e}\,\text{=}\,\text{2}\text{.71828}...$ is the base of the natural logarithm and $\gamma \,=\,0.57721...$ is the usual Euler–Mascheroni constant. This strengthens a recent theorem of Olofsson.

Keywords

11N8011N0511M45Beurling primeMertens’ theoremgeneralized primearithmetic semigroupabstract analytic number theory
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 56 , Issue 4 , 01 December 2013 , pp. 829 - 843
Copyright
Copyright © Canadian Mathematical Society 2013

References

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