Dirichlet L-series with real and complex characters and their application to solving double sums

TL;DR

This paper explores properties of Dirichlet L-series with real and complex characters, demonstrating how they can be used to express certain two-dimensional lattice sums in closed form, which were previously unsolvable.

Contribution

It introduces a method to express complex two-dimensional lattice sums in closed form using Dirichlet L-series with complex characters, expanding the analytical tools available.

Findings

Closed-form expressions for certain lattice sums achieved

Properties of L-series with complex characters elucidated

Enhanced techniques for evaluating two-dimensional sums

Abstract

A description of the properties of \L with complex characters is given. By using these, together with the more familiar \L with real characters, it is shown how certain two dimensional lattice sums, which previously could not be put into closed form, may now be expressed in this way.