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}
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:
M. N. Dobrovolskii, N. N. Dobrovolskii, N. M. Dobrovolskii, “Oblasti skhodimosti dzeta-funktsii nekotorykh monoidov naturalnykh chisel”, Chebyshevskii sb., 24:2 (2023), 154–164
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
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
N. N. Dobrovolskii, I. Yu. Rebrova, N. M. Dobrovolskii, “Entropiya dlya nekotorykh monoidov naturalnykh chisel”, Chebyshevskii sb., 23:5 (2022), 57–71
N. N. Dobrovolskii, I. Yu. Rebrova, N. M. Dobrovolskii, “Obratnaya zadacha dlya osnovnogo monoida tipa q”, Chebyshevskii sb., 23:4 (2022), 64–76
A. V. Kirilina, “O slaboi teoreme universalnosti”, Chebyshevskii sb., 21:4 (2020), 308–313
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
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
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
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
N. N. Dobrovol'sky, “One model Zeta function of the monoid of natural numbers”, Doklady Mathematics (Supplementary issues), 106:2 (2022), 192–200
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
N. N. Dobrovolskii, “Dzeta-funktsiya monoidov s zadannoi abstsissoi absolyutnoi skhodimosti”, Chebyshevskii sb., 19:2 (2018), 142–150
I. Yu. Rebrova, A. V. Kirilina, “N. M. Korobov i teoriya giperbolicheskoi dzeta-funktsii reshetok”, Chebyshevskii sb., 19:2 (2018), 341–367
N. N. Dobrovolskii, A. O. Kalinina, M. N. Dobrovolskii, N. M. Dobrovolskii, “O monoide kvadratichnykh vychetov”, Chebyshevskii sb., 19:3 (2018), 95–108