  Hardy-Littlewood-Sobolev (HLS) Inequality fails in the "critical" case:
\mu=n. However, for discrete HLS, we can derive a finite form of HLS inequality
with logarithm correction for a critical case: \mu=n and p=q, by limiting the
inequality on a finite domain. The best constant in the inequality and its
corresponding solution, the optimizer, are studied. First, we obtain a sharp
estimate for the best constant. Then for the optimizer, we prove the uniqueness
and a symmetry property. This is achieved by proving that the corresponding
Euler-Lagrange equation has a unique nontrivial nonnegative critical point.
Also, by using a discrete version of maximum principle, we prove certain
monotonicity of this optimizer.
