location:  Publications → journals
Search results

Search: All articles in the CMB digital archive with keyword abstract analytic number theory

 Expand all        Collapse all Results 1 - 1 of 1

1. CMB 2012 (vol 56 pp. 829)

Pollack, Paul
 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 theoryCategories:11N80, 11N05, 11M45
 top of page | contact us | privacy | site map |