On representations of a number as a sum of three squares

AbstractWe give a variety of results involving s(n), the number of representation of n as a sum of three squares. Using elementary techniques we prove that if 9 †n, via the theory of modular forms of half integer weight we prove the corresponding result with 3 replaced by p, an odd prime. This leads to a formula for s(n) in terms of s(n′), where n′ is the square-free part of n.We also find generating function formulae for various subsequences of {s(n)}, for instanc