Distributive and analytic properties of lattice sums

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Journal Article
Journal of Mathematical Physics, 2004, 45 (7), pp. 2560 - 2578
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We use sums over Bessel functions of the first kind to derive a convenient form of the Poisson summation identity relating sums over direct lattices in two dimensions to sums over reciprocal lattices. After three simple examples of the use of the identity, we consider sums over complex powers of the radial distance to lattice points, and also sums incorporating factors exp(4imψp) depending on angles of lattice points. We study the distribution of zeros of lattice sums, and show two which seemingly obey the Riemann hypothesis, and a third which does not. We provide a reflection formula for angular lattice sums, and a Macdonald function sum for the lowest order angular lattice sum. © 2004 American Institute of Physics.
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