
N. M. Korobov and the theory of the hyperbolic zeta function of lattices
I. Yu. Rebrova^{}, A. V. Kirilina^{} ^{} Tula State Pedagogical University
Abstract:
The paper continues the study of the role Of N. M. Korobova in the development of the numbertheoretic method in the approximate analysis. One of the Central places in the numericaltheoretical method in the approximate analysis is the method of optimal coefficients. The first example of a hyperbolic Zeta function of lattices appeared in the works of N. M. Korbova and N. S. bakhvalova in 1959 as an evaluation of integration error on the class $E_s^\alpha$ using quadrature formulas constructed on parallelepiped grids. In this paper, 5 stagesdirections in the theory of hyperbolic Zeta function of lattices are distinguished. First, it is the stage of formation of the General theory, which historically occupies the period from 1959 to 1990. During this period, the theory of quadrature formulas with generalized parallelepiped grids was constructed and it was shown that the error rate of the approximate integration on the class $E_s^\alpha$ is either equal to the hyperbolic Zeta function of lattices, the case of an integer lattice, or is estimated from above through it in the case of an arbitrary lattice. The second stage began in the mid90s, when a new direction of research of the hyperbolic Zeta function of lattices as a function of the complex argument $\alpha=\sigma+it$ on the metric space of lattices appeared. This direction continues to develop to the present time. The next stage, which also began in the mid90s, was related to the consideration of the generalized hyperbolic Zeta function of lattices, or in other words, the hyperbolic Zeta function on the folded lattices. The fourth stage, which became an independent direction of research, began in the late 90s, in the early 2000s. It is related to the question of obtaining a functional equation for the analytic continuation of the hyperbolic Zeta function of lattices. Finally, the last new direction of this theory logically emerged from the previous ones is connected with the study of Zeta functions of monoids of natural numbers. The paper reveals the defining role of Professor N. M. Korobova in the formation and development of the theory of hyperbolic Zeta function of lattices.
Keywords:
a numbertheoretic method in approximate analysis, a hyperbolic lattice Zeta function.
DOI:
https://doi.org/10.22405/222683832018192341367
Full text:
PDF file (477 kB)
References:
PDF file
HTML file
UDC:
51(091)+(092) Received: 24.04.2018 Accepted:17.08.2018
Citation:
I. Yu. Rebrova, A. V. Kirilina, “N. M. Korobov and the theory of the hyperbolic zeta function of lattices”, Chebyshevskii Sb., 19:2 (2018), 341–367
Citation in format AMSBIB
\Bibitem{RebKir18}
\by I.~Yu.~Rebrova, A.~V.~Kirilina
\paper N. M. Korobov and the theory of the hyperbolic zeta function of lattices
\jour Chebyshevskii Sb.
\yr 2018
\vol 19
\issue 2
\pages 341367
\mathnet{http://mi.mathnet.ru/cheb659}
\crossref{https://doi.org/10.22405/222683832018192341367}
\elib{https://elibrary.ru/item.asp?id=37112159}
Linking options:
http://mi.mathnet.ru/eng/cheb659 http://mi.mathnet.ru/eng/cheb/v19/i2/p341
Citing articles on Google Scholar:
Russian citations,
English citations
Related articles on Google Scholar:
Russian articles,
English articles

Number of views: 
This page:  60  Full text:  11  References:  3 
