Explicit bounds for sums of squares

For an even integer $k$, let $r_{2k}(n)$ be the number of representationsof $n$ as a sum of $2k$ squares. The quantity $r_{2k}(n)$ is approximatedby the classical singular series $\rho_{2k}(n) \asymp n^{k-1}$. Deligne'sbound on the Fourier coefficients of Hecke eigenforms gives that$r_{2k}(n) = \rho_{2k}(n)\, +\, O(d(n) n^{\frac{k-1}{2}})$. We determinethe optimal implied constant in this estimate provided thateither $k/2$ or $n$ is odd. The proof requires a delicate positivityargument involving Petersson inner products.

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Published 12 July 2012