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Determine $f(n)$ such that for all $n\geq 1$,

$$\frac{1}{\varphi (n)}=\sum_{d\vert n}\left(\frac{1}{d}\right)f\left(\frac{n}{d}\right)$$

This is not a homework question, just a question I stumbled upon.

I have tried writing $\varphi (n)$ as

$$\varphi (n)=\sum_{d\vert n}^{}{\mu(d) \frac{n}{d}},$$ where $\mu$ is the Möbius function.

I am not sure if this is the right approach, but I was stuck here.

Sincere thanks for any help!

  • 0
    Have you checked to see whether Mobius inversion gives you anything?2012-11-19
  • 0
    there is a somewhat different formula for $\frac{n}{\varphi(n)}$ at [wikipedia](http://en.wikipedia.org/wiki/Euler%27s_totient_function#Other_formulae_involving_.CF.86) (with 15 at the right and the link to Dineva's paper at the end).2012-11-19

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