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

Bateman, Paul T.; Datskovsky, Boris A.; Knopp, Marvin I.

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].

Reviewer: Tom M. Apostol (Pasadena)

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