Show $\frac{\sum_{n=1}^{\infty}\frac{1}{\phi(n)^s}}{\zeta(s)}=\prod_{p}(1-1/p^s+1/(p-1)^s)$.

Question

I want to show that the following equality and that the product is absolutely convergent and uniformally convergent on compact subsets of ${s:Re(s)>1}$.

$$\frac{\sum_{n=1}^{\infty}\frac{1}{\phi(n)^s}}{\zeta(s)}=\prod_{p}(1-\frac{1}{p^s}+\frac{1}{(p-1)^s})$$

The sum above may be expressed as a Dirichlet series, but I don't want to use that. I would like some help on how to solve this problem.