Recent progress in the study of representations of integers as sums of squares

Abstract

In this article, the authors collect the recent results concerning the representations of integers as sums of an even number of squares that are inspired by conjectures of Kac and Wakimoto. They start with a sketch of Milne's proof of two of these conjectures, and they also show an alternative route to deduce these two conjectures from Milne's determinant formulas for sums of, respectively, 4s2 or 4s(s+1) triangular numbers. This approach is inspired by Zagier's proof of the Kac-Wakimoto formulas via modular forms. The survey ends with recent conjectures of the first author and Chua. © 2005 London Mathematical Society.