Universality and antiuniversality theorems for zeta functions of monoids of natural numbers

Classes of monoids were identified for which the condition of the generalized Selberg lemma is satisfied, for which the strong Selberg–Bredikhin condition is satisfied, and for which the strengthened asymptotic law in Bredikhin form is satisfied. For these classes of monoids, new results on analytical continuation to the left of the abscissa of absolute convergence are obtained. An analogue of the main lemma of S. M. Voronin is obtained from the work on the universality of the Riemann zeta function in the case of zeta functions of a monoid for which the condition of the generalized Selberg lemma or the stronger Selberg–Bredikhin condition is satisfied.
For the class of regular Selberg–Bredikhin monoids of natural numbers, we succeeded in proving the universality theorem for the zeta function of the corresponding monoid.