Theorem

The Euler totient function(1)

Proof

We need first to prove a relationship betweenthe Mobius function and

For a fixed divisorofwe must sum over all thosein the rangewhich are multiples ofIf we writethenif and only ifHence the last sum forcan be written

Ifthe product (1) is empty. In this case the product is assigned the value 1. Suppose then thatand letbe the distinct prime divisors ofThe product can be written(2)

On the right terms such asit is understood that we take the sum of all possible products Notice also that each term on the right of (2) is of the formwhereis a divisor ofwhich is either 1 or a product of distinct primes. The numeratorisSinceifis divisible by the square of anywe have that (2) equals