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Chebyshevskii Sbornik, 2018, Volume 19, Issue 1, Pages 79–105
DOI: https://doi.org/10.22405/2226-8383-2018-19-1-79-105
(Mi cheb624)
 

This article is cited in 15 scientific papers (total in 15 papers)

On monoids of natural numbers with unique factorization into prime elements

N. N. Dobrovolsky

Tula State University
References:
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 $\beta>1$ examples of Dirichlet series with an abscissa of absolute convergence $\sigma=\frac{1}{\beta}$ are constructed. For any natural $\beta>1$ examples of a pair of zeta functions $\zeta(B|\alpha)$ and $\zeta(A_{B,\beta}|\alpha)$ with the equality $\sigma_{A_{B,\beta}}=\frac{\sigma_B}{\beta}$ 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.
Funding agency Grant number
Russian Foundation for Basic Research 16-41-710194_р_центр_а
Bibliographic databases:
Document Type: Article
UDC: 511.3
Language: Russian
Citation: N. N. Dobrovolsky, “On monoids of natural numbers with unique factorization into prime elements”, Chebyshevskii Sb., 19:1 (2018), 79–105
Citation in format AMSBIB
\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}
Linking options:
  • https://www.mathnet.ru/eng/cheb624
  • https://www.mathnet.ru/eng/cheb/v19/i1/p79
  • This publication is cited in the following 15 articles:
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
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    Abstract page:312
    Full-text PDF :102
    References:47
     
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