Question 11.12 introduction to analytic number theory

Question

I am self studying number theory from Tom M Apostol and could not solve this problem on page 247 of textbook. So, asking for help here.

Question :

I have problem in only the part where I have to prove that F(s) is to be proved non zero.

Attempt:I proved that $\sum_{n=1}^{\infty} \frac{ f(n) } { n^s} $ × $\sum_{n=1}^{\infty} \frac{ f^{-1}(n) }{ n^s} $ = 1. So, to prove that F(s) is never zero, I need to prove that $f^{-1}$ is convergent. But I am unable to do so.

Can you please tell how to complete the proof?

Thanks!!