Let's define $$ A(s) = \sum_{n=1}^{\infty} \frac{\mu(n)}{\phi(n)}n^{-s},$$ i.e. Dirichlet series generated by $\mu(n)/\phi(n)$.
I'm curious whether this Dirichlet series can be represented as other forms, maybe by Riemann zeta function.
Some formula like $ \sum_{n=1}^{\infty}\mu(n)n^{-s} = 1/\zeta(s)$.
I've been searching for this problem but I'm unable to find good results.
As the coefficients form a multiplicative function, it will have an Euler product, whose $p$-term is $$\sum_{n=0}^\infty\frac{\mu(p)}{\phi(p)}\frac1{p^{ns}} =1-\frac1{(p-1)p^s}.$$ So $$A(s)=\prod_p\left(1-\frac1{(p-1)p^s}\right).$$ I can't see very much that I can do with this...