Evaluation and regularization of phase-modulated Eisenstein series and application to double Schlömilch-type sums

In the study of periodic media, conditionally convergent series are frequently encountered and their regularization is crucial for applications. We derive an identity that regularizes two-dimensional phase-modulated Eisenstein series for all Bravais lattices, yielding physically meaningful values. We also obtain explicit forms for the phase-modulated series in terms of holomorphic Eisenstein series, enabling their closed-form evaluation for important high symmetry lattices. Results are then used to obtain representations for the related double Schlomilch-type sums, which are also given for all Bravais lattices. Finally, we treat displaced lattices of high symmetry, expressing them in terms of origin-centered lattices via geometric multi-set identities. These identities apply to all classes of two-dimensional sums, allowing sums to be evaluated over each constituent of a unit cell that possesses multiple inclusions.