Smooth manifold of one-dimensional lattices

In this paper, the foundations of the theory of smooth varieties of number-theoretic lattices are laid.
The simplest case of one-dimensional lattices is considered. In subsequent articles, we will first consider the case of one-dimensional shifted lattices, then the General case of multidimensional lattices, and finally the case of multidimensional shifted lattices.
In this paper, we define a homeomorphic mapping of the space of one-dimensional lattices to the set of all real numbers $\mathbb{R}$. Thus, it is established that the space of one-dimensional lattices $PR_1$ is locally Euclidean space of dimension $1$.
Since the metric on these spaces is not Euclidean, but is "logarithmic", unexpected results are obtained in the one-dimensional case about derivatives of the main functions, such as the lattice determinant, the hyperbolic lattice parameter, the norm minimum, the lattice Zeta function, and the hyperbolic lattice Zeta function.
The paper considers the relationship of these functions with the issues of studying the error of approximate integration over parallelepipedal grids.