Let's define , $$R_{p^m,k}(n)=\#\{(a_1,\dots,a_k)\in\mathbb{Z}^k:\sum_{i=1}^ka_i^2\le n \ \text{and} \ p^m|\sum_{i=1}^ka_i^2\}$$
what will be growth bound of $R_{p^m,k}(n)$? This can be thought as a extended version of Gauss's Circle problem.


I am interested only in the case of $k=4$ but would be happy to know in general?
Any reference will be highly helpful.

  [1]: http://mercmath.wordpress.com/2011/07/01/the-circle-method-2-elementary-estimates-for-the-singular-series-for-sums-of-5-or-more-squares/