One model Zeta function of the monoid of natural numbers

The paper studies the Zeta function $\zeta(M(p_1,p_2)|\alpha)$ of the monoid $M (p_1,p_2)$ generated by Prime numbers $p_1<p_2$ of the form $3n+2$. Next,the main monoid $M_{3,1}(p_1,p_2)\subset M(p_1,p_2)$ and the main set $ A_{3,1}(p_1,p_2)= M(p_1,p_2)\setminus M_{3,1}(p_1, p_2)$ are distinguished. For the corresponding Zeta functions, explicit finite formulas are found that give an analytic continuation on the entire complex plane except for the countable set of poles. Inverse series for these Zeta functions and functional equations are found.
The paper gives definitions of three new types of monoids of natural numbers with a unique decomposition into simple elements: monoids of degrees, Euler monoids modulo $q$ and unit monoids modulo $q$. Provided the expression of the Zeta functions using the Euler product.
The paper discusses the effect Davenport–Heilbronn Zeta-functions of monoids of natural numbers that is associated with the appearance of zeros of the Zeta-functions of terms obtained by the classes of residues modulo.
For monoids with an exponential sequence of primes, the barrier series hypothesis is proved and it is shown that the holomorphic domain of the Zeta function of such a monoid is the complex half-plane to the right of the imaginary axis.
In conclusion, topical problems with zeta-functions of monoids of natural numbers that require further investigation are considered.