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Chebyshevskii Sb., 2015, Volume 16, Issue 4, Pages 100–149 (Mi cheb438)  

This article is cited in 5 scientific papers (total in 5 papers)

Hyperbolic zeta function of lattice over quadratic field

N. M. Dobrovol'skiia, N. N. Dobrovol'skiib, V. N. Sobolevac, D. K. Sobolevc, E. I. Yushinac

a Tula State Pedagogical University
b Tula State University
c Moscow State Pedagogical University

Abstract: This work consists of two main parts.
In the first part, which presents the introduction, given a fairly comprehensive overview of the theory of the hyperbolic Zeta-function of lattices. Unlike earlier reviews is that, firstly, most of the results of the General theory particularized to two-dimensional case. This is done because the main goal of this lattice is quadratic fields. And these lattices are two-dimensional.
Secondly, the first explicit form of the functional equation for hyperbolic Zeta-function of one and two diagonal lattices.
In the second part we investigate the behavior of the hyperbolic Zeta-function of the lattice $\Lambda(t)$ of the quadratic field when the growth parameter $t$. For applications of the theory of hyperbolic Zeta-function lattices to estimate the error of the approximate integration on the class of $E_s^\alpha$ by using generalized parallelepipedal nets with weights it is important to have assessment through growing the determinant of the lattice.
In this work, we derived a new asymptotic formula for the hyperbolic Zeta function lattices of quadratic fields. The peculiarity of this formula is that it has a main two-term member and remaining a member with the assessment of incoming constants. In this formula more specific correlation between the hyperbolic Zeta function of lattices of quadratic fields and quadratic field characteristics as: the Zeta function of the Dedekind principal ideals of a quadratic field, the derivative of the Zeta-function of Dedekind principal ideals of a quadratic field, quadratic field by the regulator and the fundamental unit of the quadratic field.
Bibliography: 31 titles.

Keywords: lattice, hyperbolic zeta function of lattice, net, hyperbolic zeta function of net, quadrature formula, parallelepiped net, method of optimal coefficients.

Funding Agency Grant Number
Russian Foundation for Basic Research 11-01-00571
15-41-03263 р_центр_а

Full text: PDF file (398 kB)
References: PDF file   HTML file
UDC: 511.9
Received: 10.01.2013

Citation: N. M. Dobrovol'skii, N. N. Dobrovol'skii, V. N. Soboleva, D. K. Sobolev, E. I. Yushina, “Hyperbolic zeta function of lattice over quadratic field”, Chebyshevskii Sb., 16:4 (2015), 100–149

Citation in format AMSBIB
\by N.~M.~Dobrovol'skii, N.~N.~Dobrovol'skii, V.~N.~Soboleva, D.~K.~Sobolev, E.~I.~Yushina
\paper Hyperbolic zeta function of lattice over quadratic field
\jour Chebyshevskii Sb.
\yr 2015
\vol 16
\issue 4
\pages 100--149

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    This publication is cited in the following articles:
    1. N. M. Dobrovolskii, N. N. Dobrovolskii, V. N. Soboleva, D. K. Sobolev, L. P. Dobrovolskaya, O. E. Bocharova, “O giperbolicheskoi dzeta-funktsii Gurvitsa”, Chebyshevskii sb., 17:3 (2016), 72–105  mathnet  elib
    2. S. S. Demidov, E. A. Morozova, V. N. Chubarikov, I. Yu. Rebrova, I. N. Balaba, N. N. Dobrovolskii, N. M. Dobrovolskii, L. P. Dobrovolskaya, A. V. Rodionov, O. A. Pikhtilkova, “Teoretiko-chislovoi metod v priblizhennom analize”, Chebyshevskii sb., 18:4 (2017), 6–85  mathnet  crossref  elib
    3. I. Yu. Rebrova, A. V. Kirilina, “N. M. Korobov i teoriya giperbolicheskoi dzeta-funktsii reshetok”, Chebyshevskii sb., 19:2 (2018), 341–367  mathnet  crossref  elib
    4. N. N. Dobrovolskii, “O dvukh asimptoticheskikh formulakh v teorii giperbolicheskoi dzeta-funktsii reshetok”, Chebyshevskii sb., 19:3 (2018), 109–134  mathnet  crossref  elib
    5. A. V. Mikhlyaeva, “Priblizhenie kvadratichnykh algebraicheskikh reshetok i setok tselochislennymi reshetkami i ratsionalnymi setkami”, Chebyshevskii sb., 19:3 (2018), 241–256  mathnet  crossref  elib
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