In the 18th century, L. Euler invented a function he called sigma to study properties of whole numbers. He was interested in comparing a positive number to the sum of its positive divisors. In this problem we extend Euler's function to fractions.
Given a positive rational number (i.e., a fraction) in simplest terms a/b, we define S(a/b) to be the sum of all positive numbers of the form x/y where x is a positive divisor of a and y is a positive divisor of b. For example, the positive divisors of 4 are 1, 2 and 4 and the positive divisors of 3 are 1 and 3, so S(4/3) = 1/1 + 2/1 + 4/1 + 1/3 + 2/3 + 4/3 = 28/3.