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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 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.
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:
    1. M. N. Dobrovolskii, N. N. Dobrovolskii, N. M. Dobrovolskii, “Oblasti skhodimosti dzeta-funktsii nekotorykh monoidov naturalnykh chisel”, Chebyshevskii sb., 24:2 (2023), 154–164  mathnet  crossref
    2. M. N. Dobrovolskii, N. N. Dobrovolskii, A. V. Afonina, N. M. Dobrovolskii, I. N. Balaba, I. Yu. Rebrova, “Teoremy universalnosti i antiuniversalnosti dlya dzeta-funktsii monoidov naturalnykh chisel”, Chebyshevskii sb., 24:4 (2023), 104–136  mathnet  crossref
    3. N. N. Dobrovolskii, M. N. Dobrovolskii, N. M. Dobrovolskii, I. B. Kozhukhov, I. Yu. Rebrova, “Monoid proizvedenii dzeta-funktsii monoidov naturalnykh chisel”, Chebyshevskii sb., 23:3 (2022), 102–117  mathnet  crossref
    4. N. N. Dobrovolskii, I. Yu. Rebrova, N. M. Dobrovolskii, “Entropiya dlya nekotorykh monoidov naturalnykh chisel”, Chebyshevskii sb., 23:5 (2022), 57–71  mathnet  crossref
    5. N. N. Dobrovolskii, I. Yu. Rebrova, N. M. Dobrovolskii, “Obratnaya zadacha dlya osnovnogo monoida tipa q”, Chebyshevskii sb., 23:4 (2022), 64–76  mathnet  crossref
    6. A. V. Kirilina, “O slaboi teoreme universalnosti”, Chebyshevskii sb., 21:4 (2020), 308–313  mathnet  crossref
    7. N. N. Dobrovol'skii, I. Yu. Rebrova, N. M. Dobrovol'skii, “Inverse problem for a monoid with an exponential sequence of primes”, Doklady Mathematics (Supplementary issues), 106:2 (2022), 181–191  mathnet  crossref  crossref
    8. N. N. Dobrovolskii, M. N. Dobrovolskii, N. M. Dobrovolskii, I. N. Balaba, I. Yu. Rebrova, “Algebra ryadov Dirikhle monoida naturalnykh chisel”, Chebyshevskii sb., 20:1 (2019), 180–196  mathnet  crossref
    9. N. N. Dobrovolskii, N. M. Dobrovolskii, I. Yu. Rebrova, A. V. Rodionov, “Monoidy naturalnykh chisel v teoretiko-chislovom metode v priblizhennom analize”, Chebyshevskii sb., 20:1 (2019), 164–179  mathnet  crossref
    10. N. N. Dobrovolskii, M. N. Dobrovolskii, N. M. Dobrovolskii, “Ob odnom obobschennom eilerovom proizvedenii, zadayuschem meromorfnuyu funktsiyu na vsei kompleksnoi ploskosti”, Chebyshevskii sb., 20:2 (2019), 156–168  mathnet  crossref
    11. N. N. Dobrovol'sky, “One model Zeta function of the monoid of natural numbers”, Doklady Mathematics (Supplementary issues), 106:2 (2022), 192–200  mathnet  crossref  crossref
    12. N. N. Dobrovolskii, A. O. Kalinina, M. N. Dobrovolskii, N. M. Dobrovolskii, “O kolichestve prostykh elementov v nekotorykh monoidakh naturalnykh chisel”, Chebyshevskii sb., 19:2 (2018), 123–141  mathnet  crossref  elib
    13. N. N. Dobrovolskii, “Dzeta-funktsiya monoidov s zadannoi abstsissoi absolyutnoi skhodimosti”, Chebyshevskii sb., 19:2 (2018), 142–150  mathnet  crossref  elib
    14. I. Yu. Rebrova, A. V. Kirilina, “N. M. Korobov i teoriya giperbolicheskoi dzeta-funktsii reshetok”, Chebyshevskii sb., 19:2 (2018), 341–367  mathnet  crossref  elib
    15. N. N. Dobrovolskii, A. O. Kalinina, M. N. Dobrovolskii, N. M. Dobrovolskii, “O monoide kvadratichnykh vychetov”, Chebyshevskii sb., 19:3 (2018), 95–108  mathnet  crossref  elib
    Citing articles in Google Scholar: Russian citations, English citations
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