Universality and antiuniversality theorems for zeta functions of monoids of natural numbers
https://doi.org/10.22405/2226-8383-2023-24-4-104-136
Abstract
Classes of monoids were identified for which the condition of the generalized Selberg lemma is satisfied, for which the strong Selberg–Bredikhin condition is satisfied, and for which the strengthened asymptotic law in Bredikhin form is satisfied. For these classes of monoids, new results on analytical continuation to the left of the abscissa of absolute convergence are obtained.
An analogue of the main lemma of S. M. Voronin is obtained from the work on the universality of the Riemann zeta function in the case of zeta functions of a monoid for which the condition of the generalized Selberg lemma or the stronger Selberg–Bredikhin condition is satisfied.
For the class of regular Selberg–Bredikhin monoids of natural numbers, we succeeded in proving the universality theorem for the zeta function of the corresponding monoid.
About the Authors
Mikhail Nikolaevich Dobrovol’skii
Geophysical centre of RAS
Russian Federation
Russian Federation
candidate of candidate of physical and mathematical
sciences
Nikolai Nikolaevich Dobrovol’skii
Tula State Lev Tolstoy Pedagogical University
Russian Federation
Russian Federation
candidate of physical and mathematical sciences
Anastasia Vyacheslavovna Afonina
Tula State Lev Tolstoy Pedagogical University
Russian Federation
Russian Federation
postgraduate student
Nikolai Mikhailovich Dobrovol’skii
Tula State Lev Tolstoy Pedagogical University
Russian Federation
Russian Federation
doctor of physical and mathematical sciences, professor
Irina Nikolaevna Balaba
Tula State Lev Tolstoy Pedagogical University
Russian Federation
Russian Federation
doctor of physical and mathematical sciences, professor
Irina Yuryevna Rebrova
Tula State Lev Tolstoy Pedagogical University
Russian Federation
Russian Federation
candidate of physical and mathematical sciences, associate professor
References
1. Bredikhin, B.M., 1958, “Free numerical semigroups with power densities”, Doklady Akademii nauk SSSR, 118:5, pp. 855–857.
2. Voronin, S. M. 1975, “Theorem on the “universality” of the Riemann zeta-function”, Math. USSR Izv., vol. 9, pp. 443–453.
3. Voronin S. M., Karacuba A. A., 1994, “Dzeta-funkcija Rimana”, Izd-vo Fiz-matlit, Moskva, 376 p.
4. Gurvic A., Kurant R., 1968, “Teorija funkcij”, Izd-vo Nauka, Moskva, 618 p.
5. Demidov S. S., Morozova E. A., Chubarikov V. N., Rebrov I. Yu., Balaba I. N., Dobrovol’skii N. N., Dobrovol’skii N. M., Dobrovol’skaya L. P., Rodionov A. V., Pikhtil’kova O. A., 2017, “Number-theoretic method in approximate analysis” Chebyshevskii Sbornik vol. 18, № 4. pp. 6–85.
6. Dobrovolsky N. N., 2017, "The zeta-function is the monoid of natural numbers with unique factorization" , Chebyshevskii Sbornik, vol 18, № 4 pp. 188–208.
7. N. N. Dobrovol’skii, 2018, “On monoids of natural numbers with unique factorization into prime elements” , Chebyshevskii sbornik, vol. 19, no. 1, pp. 79–105.
8. N. N. Dobrovol’skii, 2018, “The zeta function of monoids with a given abscissa of absolute convergence” , Chebyshevskii sbornik, vol. 24, no. 4, pp. 142–150.
9. N. N. Dobrovolsky. Distribution of simple elements in some monoids of natural numbers // Math. notes (in print).
10. N. N. Dobrovol’skii, M. N. Dobrovol’skii, N. M. Dobrovol’skii, I. N. Balaba, I. Yu. Rebrova, 2018, “About “zagrobelna the series” for the zeta function of monoids with exponential sequence of simple” , Chebyshevskii sbornik, vol. 19, no. 1, pp. 106–123.
11. N. N. Dobrovol’skii, A. O. Kalinina, M. N. Dobrovol’skii, N. M. Dobrovol’skii 2018, “On the number of prime elements in certain monoids of natural numbers” , Chebyshevskii sbornik, vol. 19, no. 2, pp. 123–141.
12. N. N. Dobrovol’skii, A. O. Kalinina, M. N. Dobrovol’skii, N. M. Dobrovol’skii 2018, “On the monoid of quadratic residues” , Chebyshevskii sbornik, vol. 19, no. 3, pp. 95–108.
13. Dobrovolskaya L. P., Dobrovolsky M. N., Dobrovol’skii N. M., Dobrovolsky N. N., 2014, “On Hyperbolic Zeta Function of Lattices” , In: Continuous and Distributed Systems. Solid Mechanics and Its Applications, Vol. 211. pp. 23–62. DOI:10.1007/978-3-319-03146-0_2.
14. N. N. Dobrovol’skii, I. Yu. Rebrova, N. M. Dobrovol’skii, 2020, "Inverse problem for a monoid with an exponential sequence of Prime numbers" , Chebyshevskii sbornik, vol. 21, no. 1, pp. 165–185.
15. N. N. Dobrovol’skii, I. Yu. Rebrova, N. M. Dobrovol’skii, 2022, “Entropy for some monoids of natural numbers” , Chebyshevskii sbornik, vol. 23, no. 5, pp. 57–71.
16. A. V. Kirilina, 2020, “On the weak universality theorem” , Chebyshevskii sbornik, vol. 21, no. 4, pp. 308–313.
17. Laurinˇcikas, A., Matsumoto, K., Steuding, J. 2003, “The universality of 𝐿-functions associated with newforms”, Izv. RAN, Ser. Mat., vol. 67 (1), pp. 83–98 (in Russian) ≡ Izv. Math., vol. 67 (1), pp. 77–90.
18. Chandrasekharan K., 1974, “Vvedenie v analiticheskuju teoriju chisel”, Izd-vo Mir, Moskva, 188 p.
19. Chudakov N. G., 1947, “Introduction to the theory of 𝐿-Dirichlet functions”, M.-L.: OGIZ, — 204 p.
Review
For citations:
Dobrovol’skii M.N., Dobrovol’skii N.N., Afonina A.V., Dobrovol’skii N.M., Balaba I.N., Rebrova I.Yu. Universality and antiuniversality theorems for zeta functions of monoids of natural numbers. Chebyshevskii Sbornik. 2023;24(4):104-136. (In Russ.) https://doi.org/10.22405/2226-8383-2023-24-4-104-136
Views:
455
Email this article 