Euler’s phi function , or totient, counts how many positive integers lower than have no common divisor with .
The function was first studied by Euler, though it was Gauss who named it and established that, for any positive integer n,
where the summation is over all positive divisors d of n.
= when . (Euler 1761)
is also naturally the number of proper fractions in lowest terms with denominator . e.g. for , there are , and .