The paper continues consideration of a new class of the Dirichlet — Zeta function of monoids of natural numbers. The main task solved in this paper is to construct a monoid of natural numbers for which the Zeta function of this monoid has a given abscissa of absolute convergence.
Previously, the author solved a similar problem of constructing a set of natural numbers for which the corresponding Zeta function has a given abscissa of absolute convergence.
To solve the problem for the Zeta function of the monoid of natural numbers there are certain difficulties associated with the need to build a sequence of primes that meet certain requirements for the growth of terms.
The notion $\sigma$-sequences $\mathbb{P}_\sigma$ of primes was introduced, whose terms satisfy the inequality $n^\sigma\le p_n(n+1)^\sigma.$ With the help of a theorem of Ingham with a cubic growth of Prime numbers was able to build a $\sigma$-a sequence of primes for any $\sigma\ge3$. For the corresponding Zeta function of a monoid generated by a given $\sigma$-sequence of primes, the abscissa of absolute convergence is $\frac{1}{\sigma}$. Thus, with the help of Ingam's theorem it was possible to solve the problem for the abscissa values of absolute convergence from $0$ to $\frac{1}{3}$. For such monoids it is possible to obtain an asymptotic formula for the Prime number distribution function $\pi_{\mathbb{P}_\sigma}(x)$: $\pi_{\mathbb{P}_\sigma}(x)=x^{\frac{1}{\sigma}}+\theta(x)$, where $-2\theta(x)-1$.
To prove the existence of a monoid of natural numbers, for whose Zeta function the abscissa value of absolute convergence is from $\frac{1}{3}$ to $1$, it was necessary to use Rosser's Prime number theorem. For this purpose, the concept $\sigma$-sequences of the second kind was introduced.
In conclusion, topical problems with zeta-functions of monoids of natural numbers that require further investigation are considered.
