Explicit bounds for sums of squares

For an even integer $k$, let $r_{2k}(n)$ be the number of representations of $n$ as a sum of $2k$ squares. The quantity $r_{2k}(n)$ is appoximated by the classical singular series ${\rho }_{2k}(n)\asymp n^{k-1}$. Deligne’s bound on the Fourier coefficients of Hecke eigenforms gives that $r_{2k}(n) = \rho_{2k}(n) + O(d(n) n^{\frac{k-1}{2}})$. We determine the optimal implied constant in this estimate provided that either $k\mathrm{/}2$ or $n$ is odd. The proof requires a delicate positivity argument involving Petersson inner products.