Then (see here )
Thus, by Möbius inversion formula: (1)
Then taking exponentials in (1) your identity follows.
Paul, that was a superb answer. I really liked the proof. By the way i am just a beginner in analytic number theory and i am finding this a bit difficult as i am learning it on my own. Difficulty not in the proofs or theorems but in solving problems.
Like, i couldn never have considered the way you taken F(n). I mean how did u get the idea...
Pls help..i really want to master number theory as its such a wonderful subject....