Abstract:
The paper continues research on a new class of Dirichlet series — zeta functions of monoids of natural numbers.
The inverse Dirichlet series for zeta functions of monoids of natural numbers with unique factorization into prime elements
and for zeta-functions of sets of prime elements of monoids with unique factorization into prime elements are studied.
For any β>1 examples of Dirichlet series with an abscissa of absolute convergence σ=1β are constructed.
For any natural β>1 examples of a pair of zeta functions ζ(B|α) and ζ(AB,β|α) with the equality σAB,β=σBβ are constructed.
Various examples of monoids and corresponding zeta functions of monoids are considered.
A number of properties of the zeta functions of monoids of natural numbers with unique factorization into prime factors are obtained.
An explicit form of the inverse series to the zeta-function of the set of primes supplemented by one is found.
An explicit form of the ratio of the Riemann zeta-function to the zeta-function of the set of primes supplemented by one is found.
Nested sequences of monoids generated by primes are considered.
For the zeta-functions of these monoids the nesting principle is formulated,
which allows to transfer the results about the coefficients of one zeta-functions to the coefficients of other zeta-functions.
In this paper the general form of all monoids of natural numbers with unique factorization into prime factors was described for the first time.
In conclusion, topical problems for zeta-functions of monoids of natural numbers that require further study are considered.
Keywords:
Riemann zeta function, Dirichlet series, zeta function of monoid of natural numbers, Euler product.
\Bibitem{Dob18}
\by N.~N.~Dobrovolsky
\paper On monoids of natural numbers with unique factorization into prime elements
\jour Chebyshevskii Sb.
\yr 2018
\vol 19
\issue 1
\pages 79--105
\mathnet{http://mi.mathnet.ru/cheb624}
\crossref{https://doi.org/10.22405/2226-8383-2018-19-1-79-105}
\elib{https://elibrary.ru/item.asp?id=36312679}
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https://www.mathnet.ru/eng/cheb624
https://www.mathnet.ru/eng/cheb/v19/i1/p79
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