Zeta function regularization technique in the electrostatics context for discrete charge distributions

Spectral functions, such as the zeta functions, are widely used in quantum field theory to calculate physical quantities. In this work, we compute the electrostatic potential and field due to an infinite discrete distribution of point charges, using the zeta function regularization technique. This method allows us to remove the infinities that appear in the resulting expression. We found that the asymptotic behavior dependence of the potential and field is similar to the cases of continuous charge distribution. Finally, this exercise can be useful for graduate students to explore spectral and special functions.