Let $\varphi$ the Euler's totient. There are many known formulas to get a closed form of the Dirichlet series associated to $\varphi$ or to related functions. I didn't find one for the function $\kappa(n) = \frac{1}{\varphi(n)}$, the inverse (multiplicatively, not in the set-theoretical sense) of the totient.
This function is clearly multiplicative, so that if we denote by $K$ the Dirichlet series of $\kappa$ we have, in the convergence domain:
$$\begin{aligned} K(s) &= \prod_p \left(\sum_{e=0}^{+\infty} \frac{1}{p^{es}\varphi(p^e)}\right)\\ &=\prod_p \left(1+\sum_{e=1}^{+\infty} \frac{1}{p^{es}p^e(1-p^{-1})}\right)\\ &= \prod_p \left(1+\sum_{e=1}^{+\infty} \frac{1}{p^{e(s+1)}(1-p^{-1})}\right)\\ &= \prod_p 1+\frac{p^{-(s+1)}}{(1-p^{-(s+1)})(1-p^{-1})}\\ \end{aligned}$$
But I don't have the feeling it can be expressed in a closed form of classical functions.