The Riemann hypothesis and the zero distribution of angular lattice sums

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Journal Article
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2011, 467 (2133), pp. 2462 - 2478
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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. This journal is © 2011 The Royal Society.
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