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Sums of squares and the preservation of modularity under congruence restrictions. (English) Zbl 1040.11018

Garvan, Frank G. (ed.) et al., Symbolic computation, number theory, special functions, physics and combinatorics. Proceedings of the conference, Gainesville, FL, USA, November 11–13, 1999. Dordrecht: Kluwer Academic Publishers (ISBN 1-4020-0101-0/hbk). Dev. Math. 4, 59-71 (2001).
The number of integral solutions of the equation \[ x^2_1+ x^2_2+\cdots+ x^2_s= 8n+ s \] is denoted by \(r_s(8n+ s)\), and the number of solutions in odd integers is denoted by \(r^*_s(8n+ s)\). It is known that if \(1\leq s\leq 7\) there exists a positive constant \(c_s\) such that \(r_s(8n+ s)= c_s r^*_s(8n+ s)\) for all \(n\geq 0\). The authors show that if \(s> 7\) no constant \(c_s\) exists so that this last relation holds, even for all sufficiently large \(n\).
For the entire collection see [Zbl 0980.00026].

MSC:

11E25 Sums of squares and representations by other particular quadratic forms
11F11 Holomorphic modular forms of integral weight
11F27 Theta series; Weil representation; theta correspondences
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