Expedition 4: The Logarithmic Decomposition
Overview
[Content placeholder: How ln(ζ(s)) separates primes from prime powers. The logarithm transforms product into sum.]
The Mathematical Background
[Placeholder: Logarithm properties, Euler product, separation into prime terms and corrections]
\[\ln(\zeta(s)) = \sum_p \frac{1}{p^s} + \frac{1}{2}\sum_p \frac{1}{p^{2s}} + \frac{1}{3}\sum_p \frac{1}{p^{3s}} + \cdots\]
What This Expedition Studies
[Placeholder: Magnitude of correction terms, geometric interpretation, connection to zero locations]
Visualizations
Plots showing ln(ζ(s)) decomposition will go here
What We've Observed
- Main term Σ(1/p^s) dominates
- Correction terms are relatively small
- Logarithmic space may be more natural for understanding multiplicative structure
What We're Curious About Next
- How do correction terms affect zero locations?
- Is there a geometric interpretation of this decomposition?
[Technical content, numerical methods for computing ln(ζ), references]