  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.
