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

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 $\{\rho ={\rho }_x+\mathrm{i}{\rho }_y\}$, 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 $\Delta =1$.

19 pages, 4 figures