APPROXIMATION APPROACH IN CERTAIN PROBLEMS OF THE THEORY OF DIRICLET SERIES WITH MULTIPLICATIVE COEFFICIENTS
https://doi.org/10.22405/2226-8383-2016-17-4-124-131
Abstract
In this paper we consider a class of Dirichlet series with multiplicative coefficients which define functions holomorphic in the right half of the complex plane, and for which there are sequences of Dirichlet polynomials that converge uniformly to these functions in any rectangle within the critical strip. We call such polynomials approximating Dirichlet polynomials. We study the properties of the approximating polynomials, in particular, for those Dirichlet series, whose coefficients are determined by nonprincipal generalized characters, i.e. finite-valued numerical characters which do not vanish on almost all prime numbers and whose summatory function is bounded. These developments are interesting in connection with the problem of the analytical continuation of such Dirichlet series to the entire complex plane, which in turn is tied with the solution of a well-known Chudakov hypothesis about every generalized character being a Dirichlet character.
About the Authors
V. N. Kuznetsov
Saratov State University
Russian Federation
Russian Federation
Dr. of techical science, Professor, Head of Department of Computer Algebra and Number Theory
O. A. Matveeva
Saratov State University
Russian Federation
Russian Federation
Ph.d. in Physical Mathematical Sciences, assistant at Department of Computer Algebra and Number Theory
References
1. Kuznetsov V. N., Matveeva O. A. "On the boundary behavior of a class of Dirichlet series"Chebyshevskij sbornik, Tula, publ TPGU, 2016, vol. 17, issue 2, pp. 162–169
2. Kuznetsov V. N., Matveeva O. A. "On the boundary behavior of a class of Dirichlet series with multiplicative coefficients"Chebyshevskij sbornik, Tula, publ TPGU, 2016, vol. 17, issue 3, pp. 115–124
3. Chudakov N. G., Linnik U. V. "On a class of completely multiplicative functions"DAN SSSR, 1950, vol.74, issue 2, pp. 133–136.
4. Chudakov N. G., Rodosskij K. A. "On a generalized character"DAN SSSR, 1950, vol.74, issue 4, pp. 1137–1138.
5. Chudakov N. G., Bredichin B. M. "Usage of Parseval’s equation for estimating summation functions of characters of numerical semigroups"UMN., 1956, vol.9, issue 2, pp. 347–360.
6. Matveev V. A., Matveeva O. A. "On a behavior of the Dirichlet series with finitely valued coefficients and bounded summatory function in the critical strip"Chebyshevskij sbornik, Tula, publ TPGU, 2016, vol. 13, issue 2, pp. 106–116
7. Karatsuba A. A. "Basics of anaytic number theory". "Nauka Moscow, 1983
8. Titchmarsh E. K., "Riemann zets-function theory "I.L. Moscow, 1953
9. Glazkov V. V. "Issledovanija po teorii chisel: Mezhvuz. sb. nauch. tr., publ. SSU, 1968, vol. 2, pp. 3–40
10. Kuznetsov V. N."Analogue of the Szego theorem for a class of Dirichlet series"Math. issues, 1984, vol. 36, № 16, pp. 805–812
11. Kuznetsov V. N., Matveeva O. A. "Extended Riemann Hypothesis and zeros of functions defined by Dirichlet series with periodic coefficients"Chebyshevskij sbornik, Tula, publ TPGU, 2010, vol. 11, issue 1, pp. 188–198
12. Korotkov A. E., Matveeva O. A. "On a numerical algorithm for determining zeros of entire functions defined by Dirichlet series with periodic coefficients"Nauchnye vedomosti Belgorodskogo gosudarstvennogo universiteta, Belgorod, publ BelGU, 2011, issue 24, pp. 47–54
13. Matveeva O. A. "Approximation polynomials and the behavior of the Dirichlet L-functions on the critical band"Izvestiia Saratovskogo un-ta. Seriia «Matematika. Informatika. Mekhanika.», 2013, vol. 13, issue 4, pp. 80–84
14. Matveeva O. A. "On the zeros of Dirichlet polynomials that approximate Dirichlet L-functions in the critical band "Chebyshevskij sbornik, Tula, publ TPGU, 2013, vol. 14, issue 2, pp. 117–121
15. Matveeva O. A. "Almost periodic functions and density theorems for Dirichlet series with periodic coefficients "Materialy ХII Mezhdunarodnoj koferencii "Algebra i teorija chisel: sovremennye problemy i prilozhenija", Tula, 2014, pp. 238–239
Review
For citations:
Kuznetsov V.N., Matveeva O.A. APPROXIMATION APPROACH IN CERTAIN PROBLEMS OF THE THEORY OF DIRICLET SERIES WITH MULTIPLICATIVE COEFFICIENTS. Chebyshevskii Sbornik. 2016;17(4):124-131. (In Russ.) https://doi.org/10.22405/2226-8383-2016-17-4-124-131
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