Smooth manifold of one-dimensional lattices and shifted lattices

In the previous work, the authors laid the foundations of the theory of smooth varieties of number-theoretic lattices. The simplest case of one-dimensional lattices is considered.
This article considers the case of one-dimensional shifted lattices. First of all, we consider the construction of a metric space of shifted lattices by mapping one-dimensional shifted lattices to the space of two-dimensional lattices.
In this paper, we define a homeomorphic mapping of the space of one-dimensional shifted lattices to an infinite two-dimensional cylinder. Thus, it is established that the space of one-dimensional shifted lattices $CPR_2$ is locally a Euclidean space of dimension $2$.
Since the metric on these spaces is not Euclidean, but is "logarithmic" , unexpected results are obtained in the one-dimensional case about derivatives of basic functions, such as the determinant of the lattice, the hyperbolic lattice parameter, the norm at least, the Zeta function and lattice hyperbolic Zeta function of lattices.
The paper considers the relationship of these functions with the issues of studying the error of approximate integration over parallelepipedal grids as the determinant of the lattice, the hyperbolic lattice parameter, the norm at least, the Zeta function and lattice hyperbolic Zeta function of lattices.
Note that the geometry of metric spaces of multidimensional lattices and shifted multidimensional lattices is much more complex than the geometry of an ordinary Euclidean space. This can be seen from the paradox of nonadditivity of the length of a segment in the space of shifted one-dimensional lattices. From the presence of this paradox, it follows that there is an open problem of describing geodesic lines in the spaces of multidimensional lattices and multidimensional shifted lattices, as well as in finding a formula for the length of the arcs of lines in these spaces. Naturally, it would be interesting not only to describe these objects, but also to obtain a number-theoretic interpretation of these concepts.
A further direction of research may be the study of the analytical continuation of the hyperbolic zeta function on the spaces of lattices and multidimensional lattices. As is known, an analytical continuation of the hyperbolic zeta function of lattices is constructed for an arbitrary Cartesian lattice. Even the question of the continuity of these analytic continuations in the left half-plane on the lattice space has not been studied. All these, in our opinion, are relevant areas for further research.