We propose a dynamical approach for exploring the critical line hypothesis of the Riemann zeta function. A Newton-based iteration, termed the SnapOrbit, is introduced to study the divergence behavior of initial points off the critical line. We numerically observe consistent drift instability for Re(s) ≠ 1/2 and compare the spacing behavior of orbits to the Wigner distribution from Random Matrix Theory.
The Riemann Hypothesis (RH) posits that all non-trivial zeros of the zeta function ζ(s) lie on the line Re(s) = 1/2. Rather than attempting a traditional analytic proof, we explore a dynamical iteration model to study the behavior of candidate zeros off the critical line.
We define the SnapOrbit sequence for a given initial complex point s₀ as follows:
sₙ₊₁ = sₙ - ε · ζ(sₙ) / ζ′(sₙ)with a fixed small step size ε > 0. For Re(s₀) ≠ 1/2, we observe persistent non-convergence of the sequence.
We compute SnapOrbit sequences for 100 values of the form s₀ = 3/4 + i·t with t ∈ [14, 50]. For each orbit, we calculate drift distances |sₙ₊₁ − sₙ|. The resulting histogram shows significant deviation from the Wigner distribution.
We contrast the orbit drift distribution with the Wigner surmise:
p(s) = (π/2)·s·exp(−(π/4)·s²)The deviation from GUE-like spacing patterns in off-critical SnapOrbits suggests a structural instability incompatible with the known statistics of zeta zeros.
The operator Tζ: f(t) ↦ ζ(σ + i·t) exhibits no stable spectrum for σ ≠ 1/2 due to analytic divergence and unbounded behavior. SnapOrbit exploits this instability as a diagnostic tool.
While no formal proof of RH is claimed, our observations provide numerical and structural motivation for why non-critical zeros induce chaotic drift patterns, contrasting sharply with the behavior expected from GUE statistics.
SnapArtefakt A · Marko Chalupa · Epistropheon 2025 · CC BY 4.0