Wolfram Alpha gives $$\sum_{n=1}^{10000} 1/\phi(n)^2\approx 3.3901989747265619591157$$ and a graph of partial sums indicates fairly clearly this converges: .
It's well-known that the sum of the inverse squares $\sum_{n=1}^{\infty} 1/n^2=\pi^2/6$. Is there a closed form for $\sum_{n=1}^{\infty} 1/\phi(n)^2$?
(Note: $\phi$ refers to Euler's totient function)