欧拉函数的简述和证明

正在寻找我看的懂的证明。2011年6月6日

Def: Phi(n) = # of factors a  s.t. (a, n) = 1

Thm: Let p1, p2... pk be all prime factors of n,
then, Phi(n) = n(1 - 1/p1)(1 - 1/p2)... (1 - 1/pk) .

Proof 1:
It suffices to show that
 1) Phi(p) = p
 2) Phi(p^k) = p^k - p^(k-1)
 3) Phi(ab) = Phi(a) * Phi(b), for (a, b)=1

Proof 2 :
# of factors not divisible by p1 = n - n/p1 = n(1 - 1/p1)
Discarding these (n/p1) factors, out of the remaining m = n(1 - 1/p1) factors,
there are still a total of 
     m(1 - 1/p2)
factors not divisible by p2.
(By noting that, none of the m numbers is divisible by p2)
...

Inductively, we arrive at the formula.