What is the best known growth bound of $r_k(n)$, where $$r_k(n)=\#\{(a_1,\dots,a_k\in\mathbb{Z}^k:\sum_{i=1}^ka_i^2=n\}?$$ Please provide some reference if known. Thanks.
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asked Sep 15, 2014 at 2:37
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2$\begingroup$ Answered here: math.stackexchange.com/questions/72378/… $\endgroup$– Igor RivinCommented Sep 15, 2014 at 3:02
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As Igor Rivin said, the question was answered here for $k\geq 5$ by Greg Martin. For $n$ not divisible by $8$, the asymptotic formula described there remains valid for $k=3$ and $k=4$ as well, but the proof techniques are slightly different. Note that for $k=3$ the singular series can vanish, but only if $r_3(n)=0$. For $n$ divisible by $8$, we have $r_3(n)=r_3(n/4)$ and $r_4(n)=r_4(n/4)$, so one can reduce to the previous case. Finally, for $k=2$ there is an explicit formula based on arithmetic in Gaussian integers.
Bounding the singular series and the error term is a subtler task.
answered Sep 15, 2014 at 3:19
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1$\begingroup$ You can find an explicit, sharp bound on the error term in the cases that $k \equiv 0 \pmod{4}$ in my paper "Explicit bounds for sums of squares". $\endgroup$ Commented Sep 15, 2014 at 12:23
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$\begingroup$ @JeremyRouse: Thank you for the valuable comment! $\endgroup$ Commented Sep 15, 2014 at 12:31