Systematic investigation of two-dimensional static array sums

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Journal Article
Journal of Mathematical Physics, 2007, 48 (3)
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We discuss general properties of doubly periodic sums over the square lattice, linking phased, Bloch-type sums in the direct lattice with displaced sums in the reciprocal lattice using the Poisson summation formula. We discuss cardinal points, where the sums reduce to a single product of Dirichlet L functions, and exhibit all cardinal points for the square lattice. We introduce a new method for evaluating lattice sums and illustrate this by solving low order systems of sums of integer order 2,3,4,5. For the last case, the analytic expressions for the sums involve complementary L functions, or alternatively L functions with complex characters. © 2007 American Institute of Physics.
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