Proving an identity of the Möbius function and Euler’s totient function product Could anyone kindly help me to prove that
$$
\sum_{d|n} \mu(d) \varphi(d) = 0
$$
for all even integers $ n \geq 2 $, where $ \mu $ is the Möbius function and $ \varphi $ is Euler’s totient function? Thank you!

$$\sum_{d|n} \mu(d) \varphi(d) $$
$$=\prod_{\text{prime }p\mid n}[\mu(1)\phi(1)+\mu(p)\phi(p)+0]$$ (using the definition of Möbius function)
$$=\prod_{\text{prime }p\mid n}[1-(p-1)]$$
