Euler product of a Dirichlet function

I'm so close to this answer but just can't quite get it 100%...

Show that if is a strongly multiplicative function then the Euler product of its Dirichlet function is of the form

What I know...

Since is strongly multiplicative... It can be written as,

where p is a prime number. (1)

The Dirichlet series I'm after is of the form,

.

Now here's where I kinda get lost. I'll write down exactly what I've been writing down as my answer.

by the geometric series.

So by (1) we can write this as...

Is this right? Something about it feels wrong, like I'm not being very rigorous or that I've missed a step...