Cesaro summation by spheres of lattice sums and Madelung constants

TL;DR

This paper investigates the convergence of 3D lattice sums using expanding spheres, demonstrating that Cesaro summation ensures convergence even when traditional methods diverge, with proofs for first and second order Cesaro summation.

Contribution

It provides the first rigorous proof that Cesaro summation guarantees convergence of lattice sums on spheres, including classical Madelung constants, where standard summation diverges.

Findings

Cesaro summation ensures convergence of lattice sums on spheres.

Elementary proof for second order Cesaro convergence.

Advanced proof for first order Cesaro using Fourier series localization.

Abstract

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.