$\begingroup$

I have been playing with the series which I had been calling the 'Double Basel problem' for the past couple of hours
$$ \sum_{n=1}^{\infty} \sum_{m=1}^\infty \frac{1}{{n^2 +m^2}}. $$ After wrestling with this for awhile, I managed to generalize a result demonstrated here. This identity is: $$ \sum_{m=1}^{\infty}\frac{1}{x^2+m^2} = \frac{1}{2x}\left[ \pi \coth{\pi x} - \frac{1}{x}\right]. $$ Hence the original series becomes: $$ \sum_{n=1}^{\infty} \frac{1}{2n}\left[\pi \coth{\pi n} - \frac{1}{n} \right]. $$

I have no idea where to go next with this problem. I seriously doubt that this series is convergent; however, I have been unable to prove it.

  1. Can you prove that this series is divergent?
  2. If it converges what is its value?

Thanks so much!