2023/07/12 by Laurent Bétermin, Bétermin, Laurent, Ladislav Šamaj +3
Mathematics · Physics and Astronomy · #11E45 #Advanced Mathematical Theories and Applications #Complex Variables (math.CV) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Mathematical functions and polynomials #Number Theory (math.NT) #Statistical Mechanics and Entropy
paper · pdf · doi:10.48550/arxiv.2307.06002
openalex publication_date 2023/07/12 · openalex created_date 2023/07/14 · openalex updated_date 2026/08/01
The Riemann zeta function ζ(s):= ∑n=1∞ 1/ns can be interpreted as the energy per point of the lattice ℤ, interacting pairwisely via the Riesz potential 1/rs. Given a parameter Δ∈ (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 ℂ ∖ \ 1\. In this paper, we perform numerical investigations of the zeros of the energy \ ρ=ρx+\rm iρy\, which are defined by E(ρ,Δ)=0. The numerical results reveal that in the Riemann limit Δ→ 1- theses zeros include the anticipated critical zeros of the Riemann zeta function with \Re(ρx)=(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 -∞ as Δ→ 1-, i.e., they become invisible at the Riemann's Δ=1.