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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.

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A proof can be found here, in the artilce of Jeremy Rouse. It uses the "singular series" which gives the estimates, i.e., $r_k(n) \asymp n^{k/2-1}$, for even $k$. For more references in the literature, see here, in particular the answer of Greg Martin.

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  • $\begingroup$ Thanks for your answer. So basically, all of these proofs rely on the Hardy-Littlewood circle method + showing that the singular series is bounded. Am I correct in understanding that there is no simpler approach, even if I don't care about the values of the constants $C,c$ but only about their existence? $\endgroup$
    – Yoni Rozenshein
    Commented Feb 11, 2015 at 16:31

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