On two asymptotic formulas in the theory of hyperbolic Zeta function of lattices

N. N. Dobrovolsky

Čebyševskij sbornik, Tome 19 (2018) no. 3, pp. 109-134

Résumé

The paper considers new variants of two asymptotic formulas from the theory of hyperbolic Zeta function of lattices. First, we obtain a new asymptotic formula for the hyperbolic Zeta function of an algebraic lattice obtained by stretching $t$ times over each coordinate of a lattice consisting of complete sets of algebraically conjugate algebraic integers running through a ring of algebraic integers of a purely real algebraic field of degree $s$ for any natural $s\ge2$. Second, we obtain a new asymptotic formula for the number of points of an arbitrary lattice in a hyperbolic cross. In the first case, it is shown that the main term of the asymptotic formula for the hyperbolic Zeta function of an algebraic lattice is expressed in terms of the lattice determinant, the field controller, and the values of the Dedekind Zeta function of the principal ideals and its derivatives up to the order of $s-1$. For the first time an explicit formula of the residual term is written out and its estimation is given. In the second case, the principal term of the asymptotic formula is expressed in terms of the volume of the hyperbolic cross and the lattice determinant. An explicit form of the residual term and its refined estimate are given. In conclusion, the essence of the method of parametrized sets used in the derivation of asymptotic formulas is described.

Détail

Keywords: algebraic lattice, hyperbolic Zeta function of algebraic lattice, Dedekind Zeta function of principal ideals, hyperbolic cross, lattice points in hyperbolic cross.