The hexagonal versus the square lattice

We establish Schmutz Schaller’s conjecture that the hexagonal lattice is better' than the square lattice. Schmutz Schaller (Bulletin of the AMS 35 (1998), p. 201), motivated by considerations from hyperbolic geometry, conjectured that in dimensions 2 to 8 the best known lattice sphere packings have maximal lengths’ and goes on to write: “In dimension 2 the conjecture means in particular that the hexagonal lattice is `better’ than the square lattice. More precisely, let 0<h_1<h_2<… be the positive integers, listed in ascending order, which can be written as h_i=x^2+3y^2 for integers x and y. Let 0<q_1<q_2<… be the positive integers, listed in ascending order, which can be written as q_i=x^2+y^2 for integers x and y. Then the conjecture is that q_i<=h_i for i=1,2,3,…” Our proof requires computational prime number theory in combination with methods from a preprint of the first author (to appear in Math. Comp.), arXiv:math.NT/0112100.