# American Institute of Mathematical Sciences

doi: 10.3934/cpaa.2021153
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## Cesaro summation by spheres of lattice sums and Madelung constants

 1 Department of Mathematics, University of Surrey, GU27XH, Guildford, UK 2 School of Mathematics and Statistics, Lanzhou University, Lanzhou, 730000, China

* Corresponding author

Received  July 2021 Revised  September 2021 Early access September 2021

Fund Project: The first author is supported by the LMS URB grant 1920-04. The second author is supported by the EPSRC grant EP/P024920/1

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.

Citation: Benjamin Galbally, Sergey Zelik. Cesaro summation by spheres of lattice sums and Madelung constants. Communications on Pure & Applied Analysis, doi: 10.3934/cpaa.2021153
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A figure plotting $N$th partial sums of (3.4) with $a = 0$ and $s = \frac12$ up to N = 5000
A figure plotting $N$th partial sums of (3.15) with $a = 0$ and $s = \frac12$ up to N = 5000