[Math] simple way to compute the number of ways to write a positive integer as the sum of three squares

Question

It's a standard theorem that the number of ways to write a positive integer N as the sum of two squares is given by four times the difference between its number of divisors which are congruent to 1 mod 4 and its number of divisors which are congruent to 3 mod 4. Alternatively, there are no such representations if the prime factorization of N contains any prime of form 4k+3 an odd number of times. If the prime factorization of N contains all such primes an even number of times, then we have

r₂(N) = 4(b₁ + 1)(b₂ + 1)…(b_r+1)

where b₁, …, b_r are the exponents of the primes congruent to 1 mod 4 in the factorization of N.

For example, 325 = 5² × 13 can be written in 4(2+1)(1+1) = 24 ways as a sum of squares. These are 18² + 1², 17² + 6², 15² + 10², and the representations obtained from these by changing signs and/or permuting.

Is there an analogous formula in the three-square case? I know that an integer can be written as the sum of three squares if and only if it is not of the form 4^m(8n+7). There is a simple argument that shows that the number of ways to write all integers up to N as a sum of three squares is asymptotically 4πN^3/2/3 — representations of an integer less than N as a sum of three squares can be identified with points in the ball in R³ centered at the origin with radius N^1/2. Differentiating, a "typical" integer near N should have about 2πN^1/2 representations as a sum of three squares. From playing around with some data it looks like

lim_{n → ∞} #{ k ≤ n and r₃(k)/k^1/2 ≤ x} / n

might be a nonzero constant. That is, for each positive real x, the probability that a random integer k can be written in no more than x k^1/2 ways approaches some constant in the open interval (0, 1) as k → ∞.

One way to prove this (if it is in fact true) would be if there were some formula for r₃(k), in terms of the prime factorization, which is why I'm curious.

(I apologize if this is something that is well-known to number theorists, although I'd appreciate a pointer if it is. I am not a number theorist, I just play around with this sort of thing every so often and generate amusing conjectures.)

Answer 1

I just can't stop myself from putting up the following, from the MOTD on the Berkeley server:

Oct  2: Warning: Due to a known bug, the default Linux document viewer
        evince prints N*N copies of a PDF file when N copies requested.
        As a workaround, use Adobe Reader acroread for printing multiple
        copies of PDF documents, or use the fact that every natural number
        is a sum of at most four squares.