Zeros of Lattice Sums: 3. Reduction of the Generalised Riemann Hypothesis to Specific Geometries

TL;DR

This paper investigates the zeros of a specific lattice sum related to the Riemann zeta and Dirichlet beta functions, showing that their arrangement supports the Generalised Riemann Hypothesis under certain geometric conditions.

Contribution

It extends previous work to analyze the zero distribution of a lattice sum, linking it to the GRH and identifying geometric arrangements where the hypothesis holds.

Findings

Zeros of the lattice sum lie on the critical line or intersect specific geometric lines.

Zeros are interleaved with those of analytic functions on the critical line.

All identified arrangements support the validity of the GRH.

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). The treatment given here is an extension of that in two previous papers (arxiv:1601.01724, 1602.06330), where it was shown that non trivial zeros of the double sum either lie on the critical line or on lines of unit modulus of an analytic function intersecting the critical line. The extension enables more specific conclusions to be drawn about the arrangement of zeros of the double sum on the critical line, which are interleaved with zeros of analytic functions, all of which lie on the critical line. Possible arrangements of zeros are studied, and it is shown that in all identified cases the GRH holds.