Let 
denote the Möbius function and 
the Euler’s totient function. Prove that
![\[\frac{n}{\varphi(n)}=\sum_{d \mid n} \frac{\mu^2(d)}{\varphi(d)}\]](https://www.math.tolaso.com.gr/wp-content/ql-cache/quicklatex.com-e4497a2636d78b4be56adc3d986138a4_l3.png "Rendered by QuickLaTeX.com")
Solution
Since both functions are multiplicative it suffices to prove the identity for prime numbers. Hence for 
we have
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Let denote the Möbius function and the Euler’s totient function. Prove that
Solution
Since both functions are multiplicative it suffices to prove the identity for prime numbers. Hence for we have