Proc. R. Soc. A (2011) 467, 2462–2478 doi:10.1098/rspa.2010.0566 Published online 16 March 2011 The Riemann hypothesis and the zero distribution of angular lattice sums BY ROSS C. MCPHEDRAN 1, *, LINDSAY C. BOTTEN 2 , DOMINIC J. WILLIAMSON 1 AND NICOLAE-ALEXANDRU P. NICOROVICI 1,† 1 CUDOS, School of Physics, University of Sydney, New South Wales 2006, Australia 2 School of Mathematical Sciences, University of Technology, Sydney, New South Wales 2007 Australia We give analytical results pertaining to the distributions of zeros of a class of sums which involve complex powers of the distance to points in a two-dimensional square lattice and trigonometric functions of their angle. Let C (0, 1; s ) denote the product of the Riemann zeta function and the Catalan beta function, and let C (1, 4m; s ) denote a particular set of angular sums. We then introduce a function that is the quotient of the angular lattice sums C (1, 4m; s ) with C (0, 1; s ), and use its properties to prove that C (1, 4m; s ) obeys the Riemann hypothesis for any m if and only if C (0, 1; s ) obeys the Riemann hypothesis. We furthermore prove that if the Riemann hypothesis holds, then C (1, 4m; s ) and C (0, 1; s ) have the same distribution of zeros on the critical line (in a sense made precise in the proof). We also show that if C (0, 1; s ) obeys the Riemann hypothesis and all its zeros on the critical line have multiplicity one, then all the zeros of every C (1, 4m; s ) have multiplicity one. We give numerical results illustrating these and other results. Keywords: lattice sums; Dirichlet L functions; Riemann hypothesis 1. Introduction In this paper, we will compare the distributions of zeros of a class of sums over two-dimensional lattices involving trigonometric functions of the angle to points in the lattice, and a complex power 2s (with s = s + it ) of their distance from the lattice origin with those of a similar sum not involving trigonometric functions. The latter was evaluated analytically by Lorenz (1871) and Hardy (1920), and was shown to have the value 4z(s )L −4 (s ), where z(s ) denotes the Riemann zeta function and L −4 (s ) denotes a particular Dirichlet L function called the Catalan beta function. We will give numerical results suggesting that key properties of the distributions of zeros are the same for all the members of the set of angular sums, denoted C(1, 4m; s ), and for the angular independent sum, denoted C(0, 1; s ). We will also give proofs of *Author for correspondence (ross@physics.usyd.edu.au). † Deceased 19 December 2010. Received 30 October 2010 Accepted 7 February 2011 This journal is © 2011 The Royal Society 2462 on October 20, 2016 http://rspa.royalsocietypublishing.org/ Downloaded from on October 20, 2016 http://rspa.royalsocietypublishing.org/ Downloaded from on October 20, 2016 http://rspa.royalsocietypublishing.org/ Downloaded from on October 20, 2016 http://rspa.royalsocietypublishing.org/ Downloaded from