CARMA ANALYSIS AND NUMBER THEORY SEMINAR Speaker: Prof Ross McPhedran, The University of Sydney Title: The Riemann Hypothesis for Angular Lattice Sums Location: Room V205, Mathematics Building (Callaghan Campus) The University of Newcastle Time and Date: 3:30 pm, Wed, 11th Aug 2010 Abstract: We discuss 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. We give a general expression which permits numerical evaluation of members of the class of sums to arbitrary order. We use this to illustrate numerically the properties of trajectories along which the real and imaginary parts of the sums are zero, and we show results for the first two of a particular set of angular sums (denoted C(1, 4 m; s)) which indicate their density of zeros on the critical line of the complex exponent is the same as that for the product (denoted C(0, 1; s)) of the Riemann zeta function and the Catalan beta function. We then introduce a function which is the quotient of the angular lattice sums C(1, 4 m; s) with C(0, 1; s), and use its properties to prove that C(1, 4 m; 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 ,4 m; s) and C(0, 1; s) have the same distribution of zeros on the critical line (in a sense made precise in the proof). Download: Seminar slides (511 KB) [Permanent link]