Evidence of Long Range Order in the Riemann Zeta Function

TL;DR

This paper presents a statistical analysis of the Riemann zeta function's contour lines, revealing scale-invariant properties and consistent strip widths, suggesting long-range order in its structure.

Contribution

It provides new evidence of scale invariance and long-range order in the Riemann zeta function's contour lines through statistical analysis.

Findings

Contour lines form strips with consistent average width

Primary zero positions are scale invariant

Evidence of long-range order in the zeta function

Abstract

We have done a statistical analysis of some properties of the contour lines Im$(\zeta (s))$ = 0 of the Riemann zeta function. We find that this function is broken up into strips whose average width on the critical line does not appear to vary with height. We also compute the position of the primary zero for the lowest 200 strips, and find that this probability distribution also appears to be scale invariant.