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The paper studies the zeta function of the monoid generated by prime numbers of the form 3n + 2. The main monoid and the main set are defined. For the corresponding zeta functions, explicit finite formulas are found that give their analytic continuations to the entire complex plane, except for a countable set of poles. Inverse series for these zeta functions are found, and functional equations are derived. Three new types of monoids of natural numbers with a unique prime factorization are defined, namely, monoids of powers, Euler monoids modulo q, and unit monoids modulo q. Expressions for their zeta functions in terms of the Euler product are given. The Davenport–Heilbronn effect for zeta functions of monoids of natural numbers is discussed, which means the appearance of zeros of the zeta functions for terms obtained by decomposition into residue classes to some modulus. 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. [ABSTRACT FROM AUTHOR]

We introduce a family of rational functions R N (a , d 0 , d 1 , ... , d N) with the following property. Let d 0 , d 1 , ... , d N be equal respectively to the value of some function f (s) and the values of its first N derivatives calculated at a certain complex number a lying not too far from a zero ρ of this function. It is expected that the value of R N (a , d 0 , d 1 , ... , d N) is very close to ρ. We demonstrate this phenomenon on several numerical instances with the Riemann zeta function in the role of f (s). For example, for N = 10 and a = 0.6 + 14 i we have | R 10 (a , d 0 , d 1 , ... , d 10) − ρ 1 | < 10 − 18 where ρ 1 = 0.5 + 14.13... i is the first non-trivial zeta zero. Also we define rational functions R N , n (a , d 0 , d 1 , ... , d N) which (under the same assumptions) have values which are very close to n − ρ , that is, to the terms from the Dirichlet series for the zeta function calculated at its zero. In the case when a is between two consecutive zeros, say ρ l and ρ l + 1 , functions R N , n (a , d 0 , d 1 , ... , d N) approximate neither of n − ρ l no n − ρ l + 1 ; nevertheless, they allow us to approximate the sum n − ρ l + n − ρ l + 1 and the product n − ρ l n − ρ l + 1 and hence to calculate both n − ρ l and n − ρ l + 1 by solving corresponding quadratic equation. [ABSTRACT FROM AUTHOR]