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Recent Progress in the Study of Representations of Integers as Sums of Squares

Recent Progress in the Study of Representations of Integers as Sums of Squares 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. 2000 Mathematics Subject Classification 11E25, 11F11. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the London Mathematical Society Wiley

Recent Progress in the Study of Representations of Integers as Sums of Squares

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References (17)

Publisher
Wiley
Copyright
© London Mathematical Society
ISSN
0024-6093
eISSN
1469-2120
DOI
10.1112/S0024609305004820
Publisher site
See Article on Publisher Site

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. 2000 Mathematics Subject Classification 11E25, 11F11.

Journal

Bulletin of the London Mathematical SocietyWiley

Published: Dec 1, 2005

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