N. M. Korobov and the theory of the hyperbolic zeta function of lattices

I. Yu. Rebrova ; A. V. Kirilina

Čebyševskij sbornik, Tome 19 (2018) no. 2, pp. 341-367.

Résumé

The paper continues the study of the role Of N. M. Korobova in the development of the number-theoretic method in the approximate analysis.One of the Central places in the numerical-theoretical 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 stages-directions 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 mid-90s, 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 mid-90s, 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.

Détail

Keywords: a number-theoretic method in approximate analysis, a hyperbolic lattice Zeta function.

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