The Riemann hypothesis and the zero distribution of angular lattice sums

Abstract

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 denote the product of the Riemann zeta function and the Catalan beta function, and let denote a particular set of angular sums. We then introduce a function that is the quotient of the angular lattice sums with , and use its properties to prove that obeys the Riemann hypothesis for any m if and only if obeys the Riemann hypothesis. We furthermore prove that if the Riemann hypothesis holds, then and have the same distribution of zeros on the critical line (in a sense made precise in the proof). We also show that if obeys the Riemann hypothesis and all its zeros on the critical line have multiplicity one, then all the zeros of every have multiplicity one. We give numerical results illustrating these and other results.