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

N. N. Dobrovol'skii ; M. N. Dobrovol'skii ; N. M. Dobrovol'skii

Čebyševskij sbornik, Tome 20 (2019) no. 2, pp. 156-168.

Résumé

The paper studies the Euler product of the form $$ P_\pi(M,a(p)|\alpha)=\prod_{p\in P(M)}\left(1-\frac{a(p)}{p^{\alpha+\pi(p)}}\right)^{-1}, $$ 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.

Détail

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

Bibliographie

[1] S. M. Voronin, A. A. Karacuba, Dzeta-funkcija Rimana, Izd-vo Fizmatlit, M., 1994, 376 pp. | MR

[2] A. Gurvic, R. Kurant, Teorija funkcij, Izd-vo Nauka, M., 1968, 618 pp.

[3] S. S. Demidov, E. A. Morozova, V. N. Chubarikov, I. Yu. Rebrov, I. N. Balaba, N. N. Dobrovol'skii, N. M. Dobrovol'skii, L. P. Dobrovol'skaya, A. V. Rodionov, O. A. Pikhtil'kova, “Number-theoretic method in approximate analysis”, Chebyshevskii Sbornik, 18:4 (2017), 6–85 | MR | Zbl

[4] N. N. Dobrovolsky, “The zeta-function is the monoid of natural numbers with unique factorization”, Chebyshevskii Sbornik, 18:4 (2017), 188–208 | Zbl

[5] N. N. Dobrovol'skii, “On monoids of natural numbers with unique factorization into prime elements”, Chebyshevskii sbornik, 19:1 (2018), 79–105 | MR

[6] N. N. Dobrovol'skii, “The zeta function of monoids with a given abscissa of absolute convergence”, Chebyshevskii sbornik, 19:2 (2018), 142–150 | MR | Zbl

[7] N. N. Dobrovol'skii, “One model Zeta function of the monoid of natural numbers”, Chebyshevskii sbornik, 20:1 (2019), 148–163 | Zbl

[8] N. N. Dobrovol'skii, M. N. Dobrovol'skii, N. M. Dobrovol'skii, I. N. Balaba, I. Yu. Rebrova, “About «zagrobelna the series» for the zeta function of monoids with exponential sequence of simple”, Chebyshevskii sbornik, 19:1 (2018), 106–123 | Zbl

[9] N. N. Dobrovol'skii, A. O. Kalinina, M. N. Dobrovol'skii, N. M. Dobrovol'skii, “On the number of prime elements in certain monoids of natural numbers”, Chebyshevskii sbornik, 19:2 (2018), 123–141 | Zbl

[10] N. N. Dobrovol'skii, A. O. Kalinina, M. N. Dobrovol'skii, N. M. Dobrovol'skii, “On the monoid of quadratic residues”, Chebyshevskii sbornik, 19:3 (2018), 95–108 | Zbl

[11] N. N. Dobrovol'skii, M. N. Dobrovol'skii, N. M. Dobrovol'skii, I. N. Balaba, I. Yu. Rebrova, “Dirichlet series algebra of a monoid of natural numbers”, Chebyshevskii sbornik, 20:1 (2019), 180–196 | Zbl

[12] K. Chandrasekharan, Vvedenie v analiticheskuju teoriju chisel, Izd-vo Mir, M., 1974, 188 pp.

[13] P. L. Chebyshev, Complete works, v. I–V, Izd-vo AN SSSR, M.–L., 1944–1951 | MR

[14] P. L. Chebyshev, Selected works, Izd-vo AN SSSR, M., 1955, 926 pp. | MR

[15] N. G. Chudakov, Introduction to the theory of $L$-Dirichlet functions, OGIZ, M.–L., 1947, 204 pp. | MR

[16] B. V. Shabat, Introduction to complex analysis, Science, M., 1969, 576 pp.