Let g(n) be the convolution inverse of Euler's totient function $\varphi(n)$. Let $n=p_1^{a_1}...p_t^{a_t}$, where $p_j$ are the distinct prime divisors of $n$. Find a formula for $g(n)$ and prove that your formula is correct.
I know that the Dirichlet convolution inverse is of the form $\varphi*g=\varepsilon$ where $\varepsilon$ is the multiplicative identity function, and that the Dirichlet convolution inverse, $g(n)$ can be found by either $$g(n)=-\frac{1}{\varphi(1)}\sum_{d|n}\varphi(d)g\left(\frac{n}{d}\right)$$or by using the Dirichlet generating functions $$F_g(s)=F_{\varphi}(s)^{-1}=\left(\frac{\zeta(s-1)}{\zeta(s)}\right)^{-1}=\frac{\zeta(s)}{\zeta(s-1)}$$ Where $\zeta$ is the Riemann zeta function. Can this final Dirichlet generating function be simplified and how would I then convert that into the function $g(n)$?