Properties of the Beurling generalized prime

In this paper we prove a generalization of Mertens ’ theorem to Beurling primes, namely that limnÑ8 1 lnn p¤n 1 p1 1 Aeγ, where γ is Euler’s constant and Ax is the asymptotic number of generalized integers less than x. Thus the limitM limnÑ8 p¤n p1 lnplnnq exists. We also show that this limit coincides with limαÑ