Let $r_k(n)$ be the number of ways to write $n$ as the sum of $k$ squares of integers.
Theorem: If $k \ge 5$ then there are constants $C,c>0$ such that for any $n$, $$cn^{k/2 - 1} \le r_k(n) \le Cn^{k/2 - 1}$$
What is the simplest way to prove it as stated? I am especially interested in the lower bound. Is there a recommended book on the subject?
I have read about the Hardy-Littlewood circle method in E. Grosswald's Representations of Integers as Sums of Squares, but it seems to give a much more precise formula than what I need, and I feel there's a chance it might be simpler than that.