On Mertens' Theorem for Beurling Primes

Abstract

Abstract.Let 1 < p1 ≤ p2 ≤ p3 ≤ … be an infinite sequence ℘ of real numbers for which pi → 1, and associate with this sequence the Beurling zeta function . Suppose that for some constant A > 0, we have . We prove that ℘ satisfies an analogue of a classical theorem of Mertens: . Here e = 2.71828 … is the base of the natural logarithm and γ= 0.57721 … is the usual Euler–Mascheroni constant. This strengthens a recent theorem of Olofsson.