This material has been published in Bull. London Math. Soc. 37 (2005), 818-826, the only definitive repository of the content that has been certified and accepted after peer review. Copyright and all rights therein are retained by Cambridge University Press. This material may not be copied or reposted without explicit permission.

Heng Huat Chan and Christian Krattenthaler

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

(11 pages)

Abstract. In this article, we 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. We start with a sketch of Milne's proof of two of these conjectures. We also show an alternative route to deduce these two conjectures from Milne's determinant formulas for sums of 4s2, respectively 4s(s+1), triangular numbers. This approach is inspired by Zagier's proof of the Kac-Wakimoto formulas via modular forms. We end the survey with recent conjectures of the first author and Chua.

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