RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
General information
Latest issue
Archive

Search papers
Search references

RSS
Latest issue
Current issues
Archive issues
What is RSS



Chebyshevskii Sb.:
Year:
Volume:
Issue:
Page:
Find






Personal entry:
Login:
Password:
Save password
Enter
Forgotten password?
Register


Chebyshevskii Sb., 2019, Volume 20, Issue 2, Pages 156–168 (Mi cheb759)  

On a generalized Eulerian product defining a meromorphic function on the whole complex plane

N. N. Dobrovol'skiia, M. N. Dobrovol'skiib, N. M. Dobrovol'skiic

a Tula State University (Tula)
b Geophysical centre of RAS (Moscow)
c Tula State L. N. Tolstoy Pedagogical University (Tula)

Abstract: The paper studies the Euler product of the form
$$ P_\pi(M,a(p)|\alpha)=\prod_{p\in P(M)}(1-\frac{a(p)}{p^{\alpha+\pi(p)}})^{-1}, $$
where $M$ is an arbitrary monoid of natural numbers formed by the set of primes $P(M)$.
Another object of study is the Dirichlet series of the form
$$ f_\pi(M|\alpha)=\sum_{n\in M}\frac{1}{n^{\alpha +\pi(n)}}. $$

It turns out that they have completely different properties. The Dirichlet series $f_\pi (M| \alpha)$ defines a holomorphic function on the entire complex plane.
And the Euler product $P_\pi(M| \alpha)$ for a monoid $M$ whose set of primes $P(M)$ is infinite, sets on the entire complex plane a meromorphic function that has a countable set of special vertical lines, each of which has a countable set of poles.
In conclusion, the relevant problem of the zeros of the function $f_\pi(M|\alpha)$ is considered.

Keywords: Riemann zeta function, Dirichlet series, zeta function of the monoid of natural numbers, Euler product.

Funding Agency Grant Number
Russian Foundation for Basic Research 19-41-710004__
This work was prepared under a grant from the RFBR № 19-41-710004 _r_.


DOI: https://doi.org/10.22405/2226-8383-2018-20-2-156-168

Full text: PDF file (700 kB)

UDC: 511.3
Received: 18.05.2019
Accepted:12.07.2019

Citation: N. N. Dobrovol'skii, M. N. Dobrovol'skii, N. M. Dobrovol'skii, “On a generalized Eulerian product defining a meromorphic function on the whole complex plane”, Chebyshevskii Sb., 20:2 (2019), 156–168

Citation in format AMSBIB
\Bibitem{DobDobDob19}
\by N.~N.~Dobrovol'skii, M.~N.~Dobrovol'skii, N.~M.~Dobrovol'skii
\paper On a generalized Eulerian product defining a meromorphic function on the whole complex plane
\jour Chebyshevskii Sb.
\yr 2019
\vol 20
\issue 2
\pages 156--168
\mathnet{http://mi.mathnet.ru/cheb759}
\crossref{https://doi.org/10.22405/2226-8383-2018-20-2-156-168}


Linking options:
  • http://mi.mathnet.ru/eng/cheb759
  • http://mi.mathnet.ru/eng/cheb/v20/i2/p156

    SHARE: VKontakte.ru FaceBook Twitter Mail.ru Livejournal Memori.ru


    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles
  • Number of views:
    This page:54
    Full text:4

     
    Contact us:
     Terms of Use  Registration  Logotypes © Steklov Mathematical Institute RAS, 2020