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1. CMB 2012 (vol 56 pp. 829)
On Mertens' Theorem for Beurling Primes 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 Categories:11N80, 11N05, 11M45 |