Generation of off-critical zeros for hypercubic Epstein zeta-functions

TL;DR

This paper explores the zeros of the Epstein zeta-function on hypercubic lattices, revealing complex structures of critical and off-critical zeros, their behavior across dimensions, and deriving new analytical results.

Contribution

It introduces a continuous dimension extension of the Epstein zeta-function and uncovers the detailed structure and distribution of its zeros, including off-critical zeros and their relation to dimension.

Findings

Critical zeros form curves enclosing regions in the complex plane.

Off-critical zeros appear in conjugate pairs tending to boundaries as dimension increases.

Exact limits and distributions of zeros are derived for extreme dimensions.

Abstract

We study the Epstein zeta-function formulated on the $d$-dimensional hypercubic lattice, $\zeta^{(d)}(s) = (1/2)\sideset{}{'}\sum_{n_1,\ldots,n_d} (n_1^2+\cdots+n_d^2)^{-s/2}$ where $\Re(s)>d$ and the summation runs over all integers excluding the origin. An analytical continuation of the Epstein function to the whole complex $s$-plane is constructed for spatial dimension $d$ being a continuous variable ranging from $0$ to $\infty$. We are interested in zeros $\rho=\rho_x+{\rm i}\rho_y$ defined by $\zeta^{(d)}(\rho) = 0$. Besides the trivial zeros, there exist "critical" zeros (on the critical line) with $\rho_x=\frac{d}{2}$ and "off-critical" zeros (off the critical line) with $\rho_x \ne \frac{d}{2}$. Our numerical results reveal that critical zeros form closed or semi-open curves $\rho_y(d)$ which enclose disjunctive regions of the complex plane $(\rho_x= d/2,\rho_y)$. Each curve involves a number of left/right edge points $\rho^*$, defined by an infinite tangent ${\rm d}\rho_y/{\rm d}d\vert_{\rho^*}$, which give rise to two conjugate tails of off-critical zeros with continuously varying dimension $d$. The curves of critical and off-critical zeros exhibit a singular expansion around edge points whose derivation resembles to the one around a critical point of mean-field type (with exponent $1/2$ for the order parameter) in many-body statistical models. Further it turns out that for each $d>9.24555\ldots$ there exists a conjugate pair of {\em real} off-critical zeros which tend to the boundaries $0$ and $d$ of the critical strip in the limit $d\to\infty$. As a by-product of the formalism, we derive an exact result for $\lim_{d\to 0} \zeta^{(d)}(s)/d$ and an equidistant distribution of critical zeros along the imaginary axis in the limit $d\to\infty$.