Thread: Inverse of Euler's Phi Function

1. Inverse of Euler's Phi Function

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.

Book Download: 4shared.com - document sharing - download Introduction To Analytic Number Theory - Apostol.pdf

2. Originally Posted by streethot
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.

Book Download: 4shared.com - document sharing - download Introduction To Analytic Number Theory - Apostol.pdf

Ah, this is a nice example why it is not a good idea to read in the middle of the a book without knowing what's been treated before: they're talking there (page 37 in Apostol's book ) about inverses wrt to Dirichlet's Multiplication, NOT usual inverses of bijective functions (which should be obvious also because the Euler's Totient Functions is far from being bijective).
Begin at page 29 reading about Dirichlet's product and its main properties. It's very beautiful stuff.

Tonio