Explicit bounds for sums of squares

Abstract

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 $k/2$ or $n$ is odd. The proof requires a delicate positivity argument involving Petersson inner products.

21 pages