On off-critical zeros of lattice energies in the neighborhood of the Riemann zeta function

TL;DR

This paper investigates the zeros of a generalized lattice energy function related to the Riemann zeta function, revealing both critical and unexpected off-critical zeros that behave distinctly as the lattice parameter approaches the Riemann limit.

Contribution

It introduces a generalized lattice energy model involving the Riemann zeta function and numerically analyzes its zeros, discovering novel off-critical zeros with specific asymptotic behaviors.

Findings

Zeros include critical zeros on the critical line as well as off-critical zeros.

Off-critical zeros have equidistant imaginary parts and diverge logarithmically in real part.

Off-critical zeros become invisible at the Riemann limit $ ho o 1^-$.

Abstract

The Riemann zeta function $\zeta(s):= \sum_{n=1}^{\infty} 1/n^s$ can be interpreted as the energy per point of the lattice $\mathbb{Z}$, interacting pairwisely via the Riesz potential $1/r^s$. Given a parameter $\Delta\in (0,1]$, this physical model is generalized by considering the energy per point $E(s,\Delta)$ of a periodic one-dimensional lattice alternating the distances between the nearest-neighbour particles as $2/(1+\Delta)$ and $2\Delta/(1+\Delta)$, keeping the lattice density equal to one independently of $\Delta$. This energy trivially satisfies $E(s,1)=\zeta(s)$ at $\Delta=1$, it can be easily expressed as a combination of the Riemann and Hurwitz zeta functions, and extended analytically to the punctured $s$-plane $\mathbb{C} \setminus \{ 1\}$. In this paper, we perform numerical investigations of the zeros of the energy $\{ \rho=\rho_x+{\rm i}\rho_y\}$, which are defined by $E(\rho,\Delta)=0$. The numerical results reveal that in the Riemann limit $\Delta\to 1^-$ theses zeros include the anticipated critical zeros of the Riemann zeta function with $\Re(\rho_x)=\frac{1}{2}$ as well as an unexpected -- comparing to the Riemann Hypothesis -- infinite series of off-critical zeros. The analytic treatment of these off-critical zeros shows that their imaginary components are equidistant and their real components diverge logarithmically to $-\infty$ as $\Delta\to 1^-$, i.e., they become invisible at the Riemann's $\Delta=1$.