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

Abstract

The Riemann zeta function $ζ(s):= \sum_{n=1}^{\infty} 1/n^s$ 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 $Δ\in (0,1]$, this physical model is generalized by considering the energy per point $E(s,Δ)$ of a periodic one-dimensional lattice alternating the distances between the nearest-neighbour particles as $2/(1+Δ)$ and $2Δ/(1+Δ)$, keeping the lattice density equal to one independently of $Δ$. This energy trivially satisfies $E(s,1)=ζ(s)$ at $Δ=1$, 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 $\{ ρ=ρ_x+{\rm i}ρ_y\}$, which are defined by $E(ρ,Δ)=0$. The numerical results reveal that in the Riemann limit $Δ\to 1^-$ theses zeros include the anticipated critical zeros of the Riemann zeta function with $\Re(ρ_x)=\frac{1}{2}$ 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 $Δ\to 1^-$, i.e., they become invisible at the Riemann's $Δ=1$.