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 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_ð_à This work was prepared under a grant from the RFBR № 19-41-710004 _r_à.

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

Full text: PDF file (700 kB)

UDC: 511.3
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}