格和球面的Cesaro求和与Madelung常数
Cesaro summation by spheres of lattice sums and Madelung constants
摘要：
我们研究了三维格点和通过扩张球面的收敛性。众所周知，与通过膨胀立方体求和相反，膨胀球方法可能导致形式上的发散级数（例如经典的NaCl-马德隆常数就是如此）。本文证明了这些级数在Cesaro意义下保持收敛。对于二阶Cesaro求和，我们给出了一个初等的收敛性证明，而一阶Cesaro求和的证明则更为复杂，并且是基于多维傅里叶级数的Riemann局部化.
英文摘要：
We study convergence of 3D lattice sums via expanding spheres. It is well-known that, in contrast to summation via expanding cubes, the expanding spheres method may lead to formally divergent series (this will be so e.g. for the classical NaCl-Madelung constant). In the present paper we prove that these series remain convergent in Cesaro sense. For the case of second order Cesaro summation, we present an elementary proof of convergence and the proof for first order Cesaro summation is more involved and is based on the Riemann localization for multi-dimensional Fourier series.
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