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If $r_k(n)$ is the number of representations of $n$ as a sum of $k$ squares, and $k\geq 5$, then it is known that the estimate

$\sum_{n \leq X}r_k(n)=C_k X^{\frac{k}{2}}+O(X^{\frac{k}{2}-1}\log{x})$X^{\frac{k}{2}}+O(X^{\frac{k}{2}-1})$is known, and for$k\geq 4$this is best possible, because$r_k(n) \neq O(n^{\frac{k}{2}-1})$for$k\geq 4$fixedo(n^{\frac{k}{2}-1})$.

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If $r_k(n)$ is the number of representations of $n$ as a sum of $k$ squares, then it is known that

$\sum_{n \leq X}r_k(n)=C_k X^{\frac{k}{2}}+O(X^{\frac{k}{2}-1}\log{x})$

and for $k\geq 4$ this is best possible, because $r_k(n) \neq O(n^{\frac{k}{2}-1})$ for $k\geq 4$ fixed.