Riv. Mat. Univ. Parma, Vol. 14, No. 1, 2023

Emre Alkan [a]

Mertens type formulas based on density

Pages: 113-152
Received: 10 June 2022
Accepted in revised form: 23 August 2022
Mathematics Subject Classification: 11N05, 11M41, 11Y60.
Keywords: Mertens type formula, Mertens density, relative natural density, Dirichlet density, associated zeta function, size of semigroup.
Authors address:
[a]: Koç University, Department of Mathematics, 34450, Sarıyer, Istanbul, Turkey

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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.

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