The space of Dirichlet series to multivariate lattices and the algebra of Dirichlet series of grids, repetitive multiplication

N. V. Maksimenko

Čebyševskij sbornik, Tome 21 (2020) no. 1, pp. 233-246

Résumé

Hyperbolic Zeta functions of lattices play an important role in the numerical-theoretic method of approximate analysis. Each such hyperbolic Zeta function of the lattice is a Dirichlet series over the truncated normal spectrum of the lattice. Therefore, the problem of analytic continuation of this class of Dirichlet series arises. As shown by N. M. Dobrovolsky and his co-authors, for any Cartesian lattice, such an analytical continuation over the entire complex plane except for the point $\alpha=1$, in which the pole of order $s$ exists. The question of the existence of an analytic continuation for arbitrary lattices remains open. Therefore, it is natural to consider the set of possible Dirichlet series generated by a given lattice, and to study the properties of this functional space over the field of complex numbers. Algebraic lattices and corresponding algebraic grids entered science in 1976 in the works of K. K. Frolov. Each such lattice is a lattice repeated by multiplication, and its normal spectrum will be a monoid of natural numbers. Therefore, we can consider the algebra of Dirichlet series corresponding to this monoid of natural numbers. This setting is new and has not been seen before in the literature. The fundamental question that is associated with this statement is the following: What analytical properties do Dirichlet series have from the corresponding space and the corresponding algebra?

Détail

Keywords: Riemann zeta function, Dirichlet series, zeta function of the monoid of natural numbers, Euler product.