Zeros of Lattice Sums: 1. Zeros off the Critical Line

TL;DR

This paper investigates zeros of two-dimensional Epstein zeta sums over rectangular lattices, revealing conditions under which zeros lie off the critical line and how they evolve with lattice parameters.

Contribution

It analyzes the behavior of zeros of lattice sums as a function of lattice ratios, showing their continuous movement and identifying parameter ranges with zeros off the critical line.

Findings

Zeros can lie off the critical line for certain lattice ratios

Zeros evolve continuously as lattice parameters change

Zeros approach the critical line at second-order zeros

Abstract

Zeros of two-dimensional sums of the Epstein zeta type over rectangular lattices of the type investigated by Hejhal and Bombieri in 1987 are considered, and in particular a sum first studied by Potter and Titchmarsh in 1935. These latter proved several properties of the zeros of sums over the rectangular lattice, and commented on the fact that a particular sum had zeros off the critical line. The behaviour of one such zero is investigated as a function of the ratio of the periods $\lambda$ of the rectangular lattice, and it is shown that it evolves continuously along a trajectory which approaches the critical line, reaching it at a point which is a second-order zero of the rectangular lattice sum. It is further shown that ranges of the period ratio $\lambda$ can be so identified for which zeros of the rectangular lattice sum lie off the critical line.