1) Let's define for $n\in\mathbb{N}$ and a fixed odd prime power $p^m$ such that $p^m|n$, $$r_{p^m,k}(n)=\#\{(a_1,\dots,a_k)\in\mathbb{Z}^k:\sum_{i=1}^ka_i^2=n\},$$ How can we find an expression and growth bound for $r_{p^m,k}(n)$ and as [we have][1] for $r_k(n) =r_{1,k}(n)$ using singular series? 2) Also if, $$R_{p^m,k}(n)=\#\{(a_1,\dots,a_k)\in\mathbb{Z}^k:\sum_{i=1}^ka_i^2\le n \ \text{and} \ p^m|\sum_{i=1}^ka_i^2\}$$ 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/