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} $\zeta_{\mathcal{P}}(s):=
\prod_{i=1}^{\infty}(1p_i^{s})^{1}$. Suppose that for some constant
$A\gt 0$, we have
$\zeta_{\mathcal{P}}(s) \sim A/(s1)$, as $s\downarrow 1$. We prove that
$\mathcal{P}$ satisfies an analogue of a classical theorem of Mertens:
$\prod_{p_i \leq x}(11/p_i)^{1} \sim A \e^{\gamma} \log{x}$, as
$x\to\infty$.
Here $\e = 2.71828\ldots$ is the base of the natural logarithm and
$\gamma = 0.57721\ldots$ is the usual EulerMascheroni 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 

2. CMB 1999 (vol 42 pp. 441)
 Berrizbeitia, P.; Elliott, P. D. T. A.

Product Bases for the Rationals
A sequence of positive rationals generates a subgroup of finite
index in the multiplicative positive rationals, and group product
representations by the sequence need only a bounded number of
terms, if and only if certain related sequences have densities
uniformly bounded from below.
Categories:11N99, 11N05 
