Interplay between critical and off-critical zeros of two-dimensional Epstein zeta functions

TL;DR

This paper investigates the distribution and interplay of critical and off-critical zeros of the two-dimensional Epstein zeta function on a rectangular lattice, revealing their geometric structure and dependence on the anisotropy parameter.

Contribution

It provides a detailed numerical analysis of the zeros' distribution, introduces the concept of edge zeros, and describes how off-critical zeros form continuous curves connecting these edge zeros.

Findings

Critical zeros form curves in the (Δ, ρ_y) plane.

Off-critical zeros are systematically generated from edge zeros.

Existence of real off-critical zeros at specific Δ ranges.

Abstract

The two-dimensional Epstein zeta function formulated on a rectangular lattice with spacings $a_x=1$ and $a_y=\Delta$, $\zeta^{(2)}(s,\Delta) = \frac{1}{2} \sum_{j,k} (j^2+\Delta^2 k^2)^{-s}$ $(\Re(s)>1)$ where the sum goes over all integers except of the origin $(j,k)=(0,0)$, is studied. It can be analytically continued to the whole complex $s$-plane except for the point $s=1$. The nontrivial zeros $\{ \rho=\rho_x+{\rm i}\rho_y \}$ of the Epstein zeta function, defined by $\zeta^{(2)} (\rho,\Delta)=0$, split into ``critical'' zeros (on the critical line $\rho_x=\frac{1}{2}$) and ``off-critical'' zeros ($\rho_x\ne\frac{1}{2}$). According to the present numerical calculation, the critical zeros form open or closed curves $\rho_y(\Delta)$ in the plane $(\Delta,\rho_y)$. Two nearest critical zeros merge at special points, referred to as left/right edge zeros, which are defined by a divergent tangent ${\rm d}\rho_y/{\rm d}\Delta\vert_{\Delta^*}$. Each of these edge zeros gives rise to a continuous curve of off-critical zeros which can thus be generated systematically. As a rule, each curve of off-critical zeros joins a pair of left and right edge zeros. It is shown that in the regions of small/large values of the anisotropy parameter $\Delta$ the Epstein zeta function can be approximated adequately by a function which reveals an equidistant distribution of critical zeros along the imaginary axis in the limits $\Delta\to 0$ and $\Delta\to\infty$. It is also found that for each $\Delta\in (0,\Delta_c^*]\cup [1/\Delta_c^*,\infty)$ with $\Delta_c^*\approx 0.141733$ there exists a pair of \emph{real} off-critical zeros, their $\rho_x$ components go to the borders $0$ and $1$ of the critical region in the limits $\Delta\to 0,\infty$. As a rule, each curve of off-critical zeros joins a pair of left and right edge zeros.