Zeros of Lattice Sums: 2. A Geometry for the Generalised Riemann Hypothesis

TL;DR

This paper explores the zeros of a specific lattice sum linked to the Riemann zeta and Dirichlet beta functions, proposing geometric conditions that relate to the truth of the Generalised Riemann Hypothesis and providing numerical evidence supporting the hypothesis.

Contribution

It introduces a geometric framework involving contour intersections to characterize zeros of the lattice sum and offers new necessary and sufficient conditions for the GRH to hold.

Findings

Over 70% of zeros lie outside the 'inner islands' and are on the critical line.

Numerical evidence supports the proposed geometric conditions for zeros.

A new sufficient condition for the Riemann Hypothesis is proposed.

Abstract

The location of zeros of the basic double sum over the square lattice is studied. This sum can be represented in terms of the product of the Riemann zeta function and the Dirichlet beta function, so that the assertion that all its non-trivial zeros lie on the critical line is a particular case of the Generalised Riemann Hypothesis (GRH). It is shown that a new necessary and sufficient condition for this special case of the GRH to hold is that a particular set of equimodular and equiargument contours of a ratio of MacDonald function double sums intersect only on the critical line. It is further shown that these contours could only intersect off the critical line on the boundary of discrete regions of the complex plane called "inner islands". Numerical investigations are described related to this geometrical condition, and it is shown that for the first ten thousand zeros of both the zeta function and the beta function over 70\% of zeros lie outside the inner islands, and thus would be guaranteed to lie on the critical line by the arguments presented here. A new sufficient condition for the Riemann Hypothesis to hold is also presented.