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.