On Mertens' theorem for Beurling primes (Q2862963)

The author proves the following main result. Let \(1<p_1 \leq p_2 \leq p_3 \leq \ldots\) be an infinite sequence of real numbers, denoted by \({\mathcal P}\), such that \(p_i \to \infty\), as \(i \to \infty\). Consider the Beurling zeta function associated to \(\mathcal P\) defined by \(\zeta_{\mathcal P}(s) = \prod_{i=1}^{\infty} (1-p_i^{-s})^{-1}\). Assume that there is a constant \(A > 0\) such that \(\zeta_{\mathcal P}(s) \sim A/(s-1)\), as \(s\downarrow 1\). Then \(\prod_{p_i\leq x} (1 - 1/p_i)^{-1} \sim A e^{\gamma} \log x\), as \(x\to \infty\), which is an analogue of Mertens' classical theorem. This strengthens a result of \textit{R. Olofsson} [J. Number Theory 131, No. 1, 45--58 (2011; Zbl 1226.11104)].