Explicit bounds for sums of squares

TL;DR

This paper establishes explicit bounds for the number of representations of integers as sums of an even number of squares, refining classical estimates using advanced bounds on Fourier coefficients.

Contribution

It provides the optimal implied constant in the error term for sums of squares representations, under specific parity conditions, using a novel positivity argument.

Findings

Determined the optimal implied constant in the error estimate.

Extended bounds to cases where either k/2 or n is odd.

Utilized a positivity argument involving Petersson inner products.

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.