Some properties of three-dimensional Klein polyhedra

We study properties of three-dimensional Klein polyhedra. The main result is as follows. Let $\mathscr{L}_s(N)$ be the set of integer $s$-dimensional lattices with determinant $N$, and let $f'(\Gamma,k)$ be the set of edges $E$ of Klein polyhedra in the lattice $\Gamma$ satisfying $\#(\Gamma\cap E)=k+1$ (that is, the integer length of the edge $E$ is $k$). Then for any $k>1$,
$$ \frac{1}{\#\mathscr{L}_s(N)}\sum_{\Gamma\in\mathscr{L}_s(N)}f'(\Gamma,k)= C'_3(k)\cdot \ln^2 N+O_k(\ln N \cdot \ln\ln N), \qquad N\to \infty, $$
where $C'_3(k)$ is a positive constant depending only on $k$, and
$$ C'_3(k)=\frac{6}{\zeta(2)\zeta(3)}\cdot\frac{1}{k^3}+O\biggl(\frac{1}{k^4}\biggr). $$

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