I was reading the book Introduction to Analytic Number Theory of Apostol and found something strange. On page 37, which says that the inverse of Euler's totient function is:

$\displaystyle \varphi^{-1}(n)=\sum_{d|n}d\mu(d)$ or $\displaystyle \varphi^{-1}(n)=\prod_{p|n}(1-p)$

But look this example:

$\displaystyle \varphi(12)=4$

and

$\displaystyle \varphi^{-1}(4)\neq 12$ if we use the formula above.

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