On the Riemann property of angular lattice sums and the one-dimensional limit of two-dimensional lattice sums BY ROSS C. MCPHEDRAN 1, * , I. J. ZUCKER 2 ,LINDSAY C. BOTTEN 3 AND NICOLAE-ALEXANDRU P. NICOROVICI 1 1 CUDOS, School of Physics, University of Sydney, Sydney, New South Wales 2006, Australia 2 Department of Physics, King’s College London, The Strand, London WC2R 2LS, UK 3 School of Mathematical Sciences, University of Technology, Sydney, New South Wales 2007, Australia 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/N, 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 z(2s) if Re( s)O1/2 and either z(2sK1) or z(2K2s) 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)Z1/2. 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. Keywords: lattice sums; Dirichlet L -functions; Riemann hypothesis 1. Introduction This paper is concerned with the properties of two-dimensional lattice sums, and in particular with the distribution of their zeros, and the connection between their properties and those of one-dimensional sums. Such sums are of interest in their own right, but also arise in many areas of physics and cosmology, where they can be used in analytic continuation regularization methods ( Elizalde 2008). They are then invaluable in studies of topics as diverse as the fluctuations of the vacuum energy, the Casimir effect, quantum fields pervading the universe and the cosmological constant. The article by Elizalde and its references, in particular Proc. R. Soc. A (2008) 464, 3327–3352 doi:10.1098/rspa.2008.0230 Published online 27 August 2008 * Author for correspondence (ross@physics.usyd.edu.au). Received 4 June 2008 Accepted 29 July 2008 3327 This journal is q 2008 The Royal Society on July 19, 2018 http://rspa.royalsocietypublishing.org/ Downloaded from