Focal Vector Structure in the Riemann Zeta Function
Description
During an experimental investigation into the behavior of the Riemann zeta function at a fixed imaginary part of the argument, unusual and distinct geometric dependencies were discovered. It was found that for a large number of complex arguments of the form s = σ + i t₀, with a fixed value t₀ = 31.7183, the function ζ(s) yields real values. A deeper analysis of the vectors connecting each such point s to its value ζ(s) revealed that they all intersect in reverse through a single common point s* ∈ ℂ. This point is not a known zero, singularity, or symmetry center. This publication documents the observation as a phenomenon of focal vector structure. Geometrically, this manifests as a series of similar triangles. Over 850 points out of 10,000 tested arguments exhibit this behavior with high precision (tolerance < 1 degree). It is unclear whether this is part of a known symmetry or a new property of ζ(s), but it represents a potential breakthrough in understanding the internal geometry of this fundamental function.