  The Hardy Inequality (HI) for potentials with countably many singularities of
the form $V=\sum_{k\in \mathbf{Z}}\frac{1}{|x-a_k|^2}$ is not a trivial issue.
In principle, the more singular poles are, the less the Hardy constant is: it
is well-known that in all the existing results about the HI with finite number
of singularities the best constants converge to 0 with the number $n$ of
singularities going to infinity.
  In this note we provide an example of nontrivial HI in right cylinders of
fixed radius $R>0$ in $\mathbf{R}^d$, for a potential $V$ defined above having
the singularities $\{a_k\}_{k\in \mathbf{Z}}$ uniformly distributed on the axis
of the cylinders. For this example we prove that an upper bound for the Hardy
constant is $(d-2)^2/4$, the clasical Hardy constant in $\mathbf{R}^d$
corresponding to one singular potential. We also prove positive lower bounds of
the Hardy constant which allow to deduce that the asymptotic behavior as $R\to
0$ of the Hardy constant coincides with $(d-2)^2/4$. The proof of the main
result lies on using a nice identity due to Allegretto and Huang (Theorems 1.1,
2.1 in reference [1]) for particularly well chosen test functions.
