Let $r_k(n)$ denote the number of representations of $n$ as a sum of $k$ squares of integers.

Suppose I know that $r_4(n) = 8\sum_{4 \nmid d \mid n} d$ (Jacobi's Theorem). Is there a way to deduce from this equality (possibly with additional $r_{2k}(n)$'s, $k>1$) information about $r_2(n)$? I mean, we have the convolution identity $r_4(n) = \sum_{i+j=n} r_2(i)r_2(j)$, so in theory $r_2$ can be recovered.

Specifically, can we deduce $\sum_{i \le n} r_2(i) = \pi n + O(\sqrt{n})$ by direct means given $r_4(n)$?

I know this idea is somewhat "backwards", but that's exactly what interests me.


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