On Mertens' Theorem for Beurling Primes

Pollack, Paul

Canadian mathematical bulletin, Tome 56 (2013) no. 4, pp. 829-843

Résumé

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.

Détail

DOI : 10.4153/CMB-2012-004-8

Mots-clés : 11N80, 11N05, 11M45, Beurling prime, Mertens’ theorem, generalized prime, arithmetic semigroup, abstract analyticnumber theory