On hyperbolic Hurwitz zeta function

N. M. Dobrovolsky ; N. N. Dobrovolsky ; V. N. Soboleva ; D. K. Sobolev ; L. P. Dobrovol'skaya ; O. E. Bocharova

Čebyševskij sbornik, Tome 17 (2016) no. 3, pp. 72-105

Résumé

The paper deals with a new object of study — hyperbolic Hurwitz zeta function, which is given in the right $\alpha$-semiplane $ \alpha = \sigma + it $, $ \sigma> 1 $ by the equality $$ \zeta_H(\alpha; d, b) = \sum_{m \in \mathbb Z} \left(\, \overline{dm + b} \, \right)^{-\alpha}, $$ where $ d \neq0 $ and $ b $ — any real number. Hyperbolic Hurwitz zeta function $ \zeta_H (\alpha; d, b) $, when $ \left\| \frac {b} {d} \right\|> 0 $ coincides with the hyperbolic zeta function of shifted one-dimensional lattice $ \zeta_H (\Lambda (d, b) | \alpha) $. The importance of this class of one-dimensional lattices is due to the fact that each Cartesian lattice is represented as a union of a finite number of Cartesian products of one-dimensional shifted lattices of the form $ \Lambda (d, b) = d \mathbb{Z} + b $. Cartesian products of one-dimensional shifted lattices are in substance shifted diagonal lattices, for which in this paper the simplest form of a functional equation for the hyperbolic zeta function of such lattices is given. The connection of the hyperbolic Hurwitz zeta function with the Hurwitz zeta function $ \zeta^* (\alpha; b)$ periodized by parameter $b$ and with the ordinary Hurwitz zeta function $ \zeta (\alpha; b) $ is studied. New integral representations for these zeta functions and an analytic continuation to the left of the line $ \alpha = 1 + it $ are obtained. All considered hyperbolic zeta functions of lattices form an important class of Dirichlet series directly related to the development of the number-theoretical method in approximate analysis. For the study of such series the use of Abel's theorem is efficient, which gives an integral representation through improper integrals. Integration by parts of these improper integrals leads to improper integrals with Bernoulli polynomials, which are also studied in this paper. Bibliography: 34 titles.

Détail

Keywords: Hurwitz zeta function, periodised Hurwitz zeta function, Hurwitz zeta function of the second kind, hyperbolic Hurwitz zeta function, lattice, hyperbolic zeta function of lattice, zeta function of lattice, Bernoulli polynomials, Hankel contour.