Expedition 1: The Polar Landscape of Zeta Zeros

Overview

[Content placeholder: What representation of primes we're exploring. Why it might be interesting. What question we have in mind.]

Motivation: zeros almost purely imaginary. We explore amplitude and phase patterns using polar coordinates.

The Mathematical Background

[Content placeholder: Plain language explanation of complex numbers, polar form vs Cartesian form. Short paragraphs with diagrams.]

Example content structure:

• Complex numbers basics
• Cartesian form: ρ = σ + it
• Polar form: ρ = r·e^(iθ)
• For zeros on critical line: σ = 1/2, so r ≈ t and θ ≈ arctan(2t) → 90°

What This Expedition Studies

[Content placeholder: Specific focus on amplitude |ρₙ| and phase arg(ρₙ) properties. Patterns in phase deviation from 90°. Resonances between zeros.]

Visualizations

[Placeholder for plots: Polar scatter plots showing (|ρₙ|, arg(ρₙ)), phase convergence to 90°, amplitude vs index]

Caption: Description in everyday language of what the visualization shows.

What We've Observed

[Content placeholder: Short summary of findings so far.]

• All zeros have arg(ρ) ≈ 88-90°
• Phase approaches 90° as magnitude increases
• Amplitude |ρₙ| grows approximately linearly with index
• [More observations to come from actual analysis]

What We're Curious About Next

[Content placeholder: Open questions, hypotheses, future experiments]

• Do amplitude ratios |ρₙ₊₁|/|ρₙ| encode information?
• Are there resonances or patterns in phase differences?
• How does this connect to the Random Matrix Theory predictions?