Mertens type formulas based on density

Abstract

We introduce a new density among sets of prime numbers which is called the Mertens density. Building on the works of Olofsson, Pollack and Wirsing, it is shown, in complete contrast with the cases of relative natural density and Dirichlet density, that the existence of Mertens density of a set of prime numbers turns out to be equivalent to Mertens type formulae and the limiting behaviors of the associated zeta function at one together with the size of the corresponding semigroup, all formed according to the underlying set of primes. Various constants, such as the Meissel-Mertens constants, appearing in the equivalent statements are shown to be related with each other through elementary formulas. This allows us to study specific partitioning properties between sets of primes taking into account their density and the asymptotics of the generated semigroup. It is further demonstrated that the Mertens density neither implies nor is implied by the relative natural density. Assuming explicit forms of the error terms, sharper versions of some of our results are also obtained.