Angular Lattice Sums and the Riemann Hypothesis

Prof Ross C. McPhedran (University of Sydney)

Title: Angular Lattice Sums and the Riemann Hypothesis

We will take Prof McPhedran to lunch before his talk, meeting at 1 PM on the 2nd floor of Carslaw and proceeding to the Grandstand.

Talk: Carslaw 175, 2:30 PM

Abstract: I will discuss how angular lattice sums arise in physical problems, and why their properties may be of interest to mathematicians. Specifically, I will identify a class of such sums which have been little investigated, but for which strong results may be proved. These sums are over the two-dimensional lattice of positive and negative integers, and involve complex powers of the distance and trigonometric functions of their angle. They are denoted C(1,4 m;s), where s is a complex variable and m is an integer. For m=0, the result C(1,0;s) has long been known in analytic form. We can prove by a relatively simple argument that all the C(1,4 m;s) for any m obey the Riemann hypothesis if and only if C(1,0;s) does. Furthermore, under this assumption, all the C(1,4 m;s) have the same distribution function for their zeros on the critical line as does C(1,0;s) .

Ross C. McPhedran, Lindsay C. Botten, Dominic J. Williamson and Nicolae-Alexandru P. Nicorovici

CUDOS, School of Physics, University of Sydney, NSW 2006, Australia, School of Mathematical Sciences, University of Technology, Sydney, N.S.W. 2007 Australia