nt.number theory - Groups related to sum of squares function?

For general odd $k=2m+1$ one can still compute the value of the singular series as Hardy does and obtain a formula similar to those for $k=5,7$, involving the value of an $L$-series with quadratic character at $s=m$. A relation to special groups as in the case $k=3$ is not visible from this, just a relation to the arithmetic of quadratic number fields.

For $k ge 9$, however, the genus of the sum of $k$ squares contains more than one integral equivalence class and by Siegel's Massformel (mass formula) evaluation of the singular series gives the average of the representation numbers for the equivalence classes in the genus and not the representation number of the individual form. (The genus of an integral quadratic form $q$ consists of those forms which have the same signature and are integrally equivalent modulo $m$ for all integral $m$.) Of course this doesn't exclude the possibility of finding a closed formula by other means, as was the case for the special dimensions of the form $4m^2$ or $4(m^2+m)$ in the work of Milne mentioned in Jagy's answer. To my knowledge at present no such formula is known for an odd number $k$ of variables.

Concerning references: The standard reference for sums of squares is still Grosswald's book. Good references for more general questions concerning the arithmetic of quadratic forms are the books of B. Jones, Y. Kitaoka, O. T. O'Meara, G. Shimura and (in german) of M. Eichler and of M. Kneser.