On the Riemann property of angular lattice sums and the one-dimensional limit of two-dimensional lattice sums

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Journal Article
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2008, 464 (2100), pp. 3327 - 3352
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We consider a general class of two-dimensional lattice sums consisting of complex powers s of inverse quadratic functions. We consider two cases, one where the quadratic function is negative definite and another more restricted case where it is positive definite. In the former, we use a representation due to H. Kober, and consider the limit u → ∞, where the lattice becomes ever more elongated along one period direction (the one-dimensional limit). In the latter, we use an explicit evaluation of the sum due to Zucker and Robertson. In either case, we show that the one-dimensional limit of the sum is given in terms of ζ(2s) if Re(s)>1/2 and either ζ(2s-1) or ζ(2-2s) if Re(s)<1/2. In either case, this leads to a Riemann property of these sums in the one-dimensional limit: their zeros must lie on the critical line Re(s)=1/2. We also comment on a class of sums that involve complex powers of the distance to points in a two-dimensional square lattice and trigonometric functions of their angle. We show that certain of these sums can have their zeros on the critical line but not in a neighbourhood of it; others are identically zero on it, while still others have no zeros on it. © 2008 The Royal Society.
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