Q:

$\sum_{m,n=1}^\infty{\frac1{mn}}$

I've surfed almost half of the Internet [:-)] but didn't succeed in finding an answers:
1) Does the series in the title have a special name?
2) Does it converge or diverge? Proof appreciated
3a) If it converges what is it value?
3b) If it diverges does there exist such $n\in\mathbb{N}$ that $\sum_{a_1,a_2,...,a_n=1}^\infty\frac1{a_1a_2...a_n}$ converges?

A:

As an alternative to the comment by Jack D'Aurizio,  to show the divergence we can simply observe that
$$\sum_{m,n=1}^\infty{\frac1{mn}}\ge \sum_{\substack{n=1}\\(m=1)}^\infty{\frac1{1\cdot n}}=\sum_{n=1}^\infty{\frac1{n}}$$
and we can generalize of course
$$\sum_{a_1,a_2,...,a_n=1}^\infty\frac1{a_1a_2...a_n}\ge \sum_{\substack{a_1=1}\\(a_2,\ldots a_n=1)}^\infty{\frac1{1\cdot a_1}}=\sum_{n=1}^\infty{\frac1{n}}$$

