论文标题
富裕总和的两个几何解释
Two Geometric Interpretations of Hardy Sums
论文作者
论文摘要
如果晶格为$ \ mathbf {z}^2 $，则在三角形中找到晶格点的数量具有经典的解决方案，而三角形的顶点具有整数值坐标。我们考虑将晶格替换为$（2 \ mathbf {z}）^2 $而替换晶格时会发生什么，并给出一个明确的公式，以用耐力的总和在三角形内的晶格点数。此外，我们将强力总和的第二个几何解释作为签名的相交数字，并具有一定的指向的地理学网。使用这种几何实现，我们通过基本参数证明了对Hardy总和的广义互惠定律。
The problem of finding the number of lattice points in a triangle has a classical solution if the lattice is $\mathbf{Z}^2$ and the vertices of the triangle have integer valued coordinates. We consider what happens when we replace the lattice by $(2 \mathbf{Z})^2$ instead and give an explicit formula for the number of lattice points inside a triangle in terms of Hardy sums. Moreover, we give a second geometric interpretation of the Hardy sums as signed intersection numbers with a certain oriented net of geodesics. Using this geometric realization, we prove a generalized reciprocity law for Hardy sums by an elementary argument.
