Sums of \(4k\) squares: a polynomial approach (Q2880112)

Let \(k\) and \(n\) be positive integers and \(S_k(u)\) denote the number of representations of \(u\) as the sum of \(k\) squares. Ramanujan gave without proof a formula for \(S_n(u)\), where \(k\) is even, letter proved by Morchell using modular forms. In the case \(k\equiv 0\text{\,mod\,}4\), the authors prove Ramanujan's formula in an entirely elementary way using only simple properties of polynomials. Explicit values of \(S_k(u)\) are determined for \(\varepsilon= 4,8,12,\dots,44\) and \(48\).