The zeta-function is the monoid of natural numbers with unique factorization

In this paper we consider a new class of Dirichlet series, the zeta functions of monoids of natural numbers. The inverse Dirichlet series for the zeta function of monoids of natural numbers are studied. It is shown that the existence of an Euler product for the zeta function of a monoid is related to the uniqueness of the factorization into prime factors in this monoid.
The notion of coprime sets of natural numbers is introduced and it is shown that for such sets the multiplicativity of minimal monoids and corresponding zeta-functions of monoids takes place.
It is shown that if all prime elements of a monoid are prime numbers, then the characteristic function of the monoid is a multiplicative function and in this case the zeta function of the monoid is a generalized L-function.
Various examples of monoids and corresponding zeta functions of monoids are considered. The relation between the inversion of the zeta function of a monoid and the generalized Möbius function on a monoid as a partially ordered set is studied by means of the divisibility of natural numbers. A number of properties of the zeta functions of monoids of natural numbers with a unique decomposition into prime factors are obtained.
The paper deals with taking the logarithm of an Eulerian product as a function of a complex argument. It is shown that a continuous function that determines the value of the logarithm of an Euler product runs through all branches of the infinite-valued function of the logarithm near its pole. The corollaries on the value of a complex-valued function of a special form near a singular point are obtained. These properties imply statements about the values of the Riemann zeta function near the boundary of the region of absolute convergence.
Using Bertrand's postulate, infinite exponential sequences of prime numbers are introduced. It is shown that corresponding zeta-functions of monoids of natural numbers converge absolutely in the whole half-plane with a positive real part. Since such zeta-functions of monoids of natural numbers can be decomposed into an Euler product in the whole region of absolute convergence, they do not have zeros in the entire half-plane with a positive real part.
In conclusion, topical problems with zeta-functions of monoids of natural numbers that require further investigation are considered.