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# On Mertens' Theorem for Beurling Primes

Published:2012-03-05
Printed: Dec 2013
• Paul Pollack,
Department of Mathematics, University of British Columbia, 1984 Mathematics Road, Vancouver, BC V6T 1Z2
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## Abstract

Let $1 \lt p_1 \leq p_2 \leq p_3 \leq \dots$ be an infinite sequence $\mathcal{P}$ of real numbers for which $p_i \to \infty$, and associate to this sequence the \emph{Beurling zeta function} $\zeta_{\mathcal{P}}(s):= \prod_{i=1}^{\infty}(1-p_i^{-s})^{-1}$. Suppose that for some constant $A\gt 0$, we have $\zeta_{\mathcal{P}}(s) \sim A/(s-1)$, as $s\downarrow 1$. We prove that $\mathcal{P}$ satisfies an analogue of a classical theorem of Mertens: $\prod_{p_i \leq x}(1-1/p_i)^{-1} \sim A \e^{\gamma} \log{x}$, as $x\to\infty$. Here $\e = 2.71828\ldots$ is the base of the natural logarithm and $\gamma = 0.57721\ldots$ is the usual Euler--Mascheroni constant. This strengthens a recent theorem of Olofsson.
 Keywords: Beurling prime, Mertens' theorem, generalized prime, arithmetic semigroup, abstract analytic number theory
 MSC Classifications: 11N80 - Generalized primes and integers 11N05 - Distribution of primes 11M45 - Tauberian theorems [See also 40E05]