Consider the series $$S=\sum\limits_{m=-\infty}^\infty \sum\limits_{n=-\infty}^\infty \frac{\sin^2(2 \pi(m-n))}{(m-n)^2}.$$ Does $S$ converge, and if so what is its value?
I understand that when $m \ne n$ the sum is trivially zero. But I am unable to evaluate it generally. I believe I should get a Kronecker Delta, but am unable to prove it analytically.