HYPERBOLIC ZETA FUNCTION OF LATTICE OVER QUADRATIC FIELD

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 Zetafunction of the lattice Λ(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 Eα s 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.

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