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Chebyshevskii Sbornik, 2018, Volume 19, Issue 1, Pages 106–123
DOI: https://doi.org/10.22405/2226-8383-2018-19-1-106-123
(Mi cheb625)
 

This article is cited in 16 scientific papers (total in 16 papers)

Hypothesis about "barier series" for the zeta-functions of monoids with the exponential sequence of primes

N. N. Dobrovolskya, M. N. Dobrovolskyb, N. M. Dobrovolskyc, I. N. Balabac, I. Yu. Rebrovac

a Tula State University
b Geophysical center of RAS, Moscow
c Tula State Pedagogical University
References:
Abstract: The work continues the study of a new class of Dirichlet series — the zeta functions of monoids of natural numbers. First of all, we study in detail the zeta function $\zeta(M(q)|\alpha)$ of geometric progression $M(q)$ with initial value equal to 1 and an arbitrary natural common ratio $q>1$, which is the simplest monoid of natural numbers with a unique decomposition into prime elements of the monoid. For a meromorphic function $\zeta(M(q)|\alpha)=\frac{q^\alpha}{q^\alpha-1}$, which have poles
$$ S(M (q))=\left\{\left. \frac{2\pi i k}{\ln q}\right| k\in\mathbb{Z}\right\} $$
representations are received:
\begin{gather*} \zeta(M(q)|\alpha)=\frac{q^{\frac{\alpha}{2}}}{\alpha\ln q}\prod_{n=1}^{\infty}\left(1+\frac{\alpha^2\ln^2 q}{4\pi^2 n^2}\right)^{-1}=\frac{1}{2}+\frac{1}{\alpha\ln q}+\sum_{n=1}^{\infty}\frac{2\alpha\ln q}{\alpha^2\ln^2 q+4n^2\pi^2}= \\ =\frac{q^{\frac{\alpha}{2}}\alpha\ln q}{4\pi^2}\Gamma\left(\frac{\alpha i\ln q }{2\pi}\right)\Gamma\left(-\frac{\alpha i\ln q }{2\pi}\right). \end{gather*}

For the zeta function $\zeta(M(\vec{p})|\alpha)$ of the monoid $M(\vec{p})$ with a finite number of primes $\vec{p}=(p_1,\ldots,p_n)$ the decomposition into an infinite product is obtained
$$ \zeta(M(\vec{p})|\alpha)=\frac{P(\vec{p})^{\frac{\alpha}{2}}}{\alpha^nQ(\vec{p})}\prod_{\nu=1}^{n}\prod_{m=1}^{\infty}\left(1+\frac{\alpha^2\ln^2 p_\nu}{4\pi^2 m^2}\right)^{-1}, $$
where $P (\vec{p})=p_1\ldots p_n$, $Q (\vec{p})=\ln p_1\ldots \ln p_n$, and a functional equation is found
$$ \zeta (M (\vec{p})|-\alpha)=(-1)^n\frac{\zeta (M (\vec{p})|\alpha)}{P (\vec{p})^\alpha}. $$

For the monoid of natural numbers $M^*(\vec{p})= \mathbb{N}\cdot M^{-1} (\vec{p})$ with a unique decomposition into prime elements, which consists of natural numbers $n$ coprime with $P (\vec{p})=p_1\ldots p_n$, and for the Euler product $P (M^*(\vec{p}) / \ alpha)$, which consists of factors for all primes other than $p_1,\ldots, p_n$, a functional equation is found
$$ \zeta(M^*(\vec{p})|\alpha)=M(\vec{p},\alpha) \zeta(M^*(\vec{p})|1-\alpha), $$
where
$$ M(\vec{p},\alpha)=M(\alpha)\cdot\frac{M_1(\vec{p},\alpha)}{M_1(\vec{p},1-\alpha)}, \quad M_1(\vec{p},\alpha)=\prod_{\nu=1}^{n}\left(1-\frac{1}{p_\nu^\alpha}\right). $$

It is proved that for any infinite set of primes $\mathbb{P}_1$ there is no analytic function equal to
$$\lim\limits_{n\to\infty} \zeta(M(\vec{p}_n)|\alpha)$$
on the whole complex plane.
The hypothesis about the barrier series for any exponential set of $ PE $ prime numbers is formulated.
In conclusion, topical problems with zeta-functions of monoids of natural numbers that require further investigation are considered.
Keywords: Riemann zeta function, Dirichlet series, zeta function of monoids of natural numbers, Euler product, logarithm of the 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, M. N. Dobrovolsky, N. M. Dobrovolsky, I. N. Balaba, I. Yu. Rebrova, “Hypothesis about "barier series" for the zeta-functions of monoids with the exponential sequence of primes”, Chebyshevskii Sb., 19:1 (2018), 106–123
Citation in format AMSBIB
\Bibitem{DobDobDob18}
\by N.~N.~Dobrovolsky, M.~N.~Dobrovolsky, N.~M.~Dobrovolsky, I.~N.~Balaba, I.~Yu.~Rebrova
\paper Hypothesis about "barier series" for the zeta-functions of monoids with the exponential sequence of primes
\jour Chebyshevskii Sb.
\yr 2018
\vol 19
\issue 1
\pages 106--123
\mathnet{http://mi.mathnet.ru/cheb625}
\crossref{https://doi.org/10.22405/2226-8383-2018-19-1-106-123}
\elib{https://elibrary.ru/item.asp?id=36312680}
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  • https://www.mathnet.ru/eng/cheb/v19/i1/p106
  • This publication is cited in the following 16 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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