Euler's totient function
Euler's totient function, , is defined as the number of positive integers less than or equal to a given positive integer that are relatively prime to that integer.
To derive the formula, let us first define the prime factorization of as where the are primes. Now, we can use a PIE argument to count the number of numbers less than or equal to are are relatively prime to it.
First, let's count the complement of what we want (i.e. all the numbers less than that share a common factor with it). There are numbers less than that are divisible by . If we do the same for each and add these up, we get
We can factor out though:
But we are obviously overcounting. So we then subtract out those divisible by two of the . We continue with this PIE argument to figure out that the number of elements in the complement of what we want is
which we can factor further as
Making one small adjustment, we write this as
Given the general prime factorization of , one can compute using the formula .
For prime p, , because all numbers less than are relatively prime to it.
For relatively prime , .
In fact, we also have for any that .
For any , we have where the sum is taken over all divisors d of .