六边形和正方形晶格
The hexagonal versus the square lattice
摘要：
我们建立了Schmutz Sünier的猜想，即六角晶格比正方晶格“更好”。Schmutz Schücher（Bulletin of the AMS 35（1998），p. 201），从双曲几何的考虑出发，阐明了在2到8维中，最著名的晶格球填充具有“最大长度”，并接着写道：“在2维中，该猜想特别意味着六边形晶格比正方形晶格”更好“。更准确地说，设0<h_1<h_2<.是按升序排列的正整数，对于整数x和y，可以写成h_i=x^2+3y^2。设0<q_1<q_2<.是按升序排列的正整数，对于整数x和y，可以写成q_i=x^2+y^2。则猜想q_i<=h_i（i= 1，2，3，.）“我们的证明需要计算素数理论与第一作者预印本中的方法相结合（出现在Math. Comp.），arXiv：math.NT/0112100.
英文摘要：
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.
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