Explicit Bounds for Sums of Squares
نویسنده
چکیده
For an even integer k, let r2k(n) be the number of representations of n as a sum of 2k squares. The quantity r2k(n) is approximated by the classical singular series ρ2k(n) nk−1. Deligne’s bound on the Fourier coefficients of Hecke eigenforms gives that r2k(n) = ρ2k(n) + O(d(n)n 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.
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