The problem statement, all variables and given/known data: Recently, a group of fellow math nerds and myself stumbled upon an interesting problem. The problem is stated: "Find the average number of representations of a positive integer as the sum of two squares." Relevant equations: N = a^(2) + b^(2), where a and b can be 1 or many corresponding sets. But we are looking for the average number of sets, for all real numbers of N. The attempt at a solution: Clearly, I realize a limit will need to be taken, somehow, of all numbers N that can be expressed as the sum of two squares. Naturally, the general equation for N would be N = a^(2) + b^(2). Also, I figured that N must meet the condition that every prime number of the form 4k+3 appears an even number of times in it's prime factorization. I have no idea how to connect these two conditions, or if more must be met. If it's any help, my friend claimed that the answer is π, but had no way to prove it. Any ideas are greatly appreciated!