On the representation of integers as sums of squares (Q2782414)

A trigonometric series identity of Ramanujan is used to derive some Lambert series identities. These identities combined with some identities of Jacobi lead to a new and simple derivation of Ramanujan's formula for \(r_{24}(n)\), the number of representations of \(n\) as a sum of 24 squares. The (known) formulas for \(r_{12}(n)\), \(r_{16}(n)\) and \(r_{20}(n)\) are also proved along the way. Let \(t_k(n)\) denote the number of representations of \(n\) as a sum of \(k\) triangular numbers: applying some simple modular transformations to \(r_k(n)\), formulas for \(t_k(n)\) are proved for \(k= 12, 16, 20\) and 24.NEWLINENEWLINEFor the entire collection see [Zbl 0980.00024].