Hyperbolic zeta function of lattice over quadratic field

N. M. Dobrovol'skii ; N. N. Dobrovol'skii ; V. N. Soboleva ; D. K. Sobolev ; E. I. Yushina

Čebyševskij sbornik, Tome 16 (2015) no. 4, pp. 100-149

Résumé

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.

Détail

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