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Chapter 5: The Zeta Function as Arithmetic Mirror

5.1 Dual Representations​

The Two Forms: ζ(s)=∑n=1∞1ns=∏p prime11−p−s\zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s} = \prod_{p \text{ prime}} \frac{1}{1-p^{-s}}

Theorem 5.1: This duality is arithmetic observing itself.

Proof: Sum = counting, Product = factoring. Their equality means arithmetic is self-consistent. ∎

5.2 The Functional Equation​

Theorem 5.2: ξ(s)=ξ(1−s)\xi(s) = \xi(1-s) forces zeros to Re(s) = 1/2.

Proof: The transformation s ↦ 1-s has fixed points only at Re(s) = 1/2. Zeros must respect this symmetry or arithmetic becomes inconsistent. ∎

5.3 Information Constraints​

Theorem 5.3: Zeros must form a 1D set.

Proof: ζ(s) is determined by 1D boundary data. Cannot encode 2D zero distribution. By symmetry, the 1D set is Re(s) = 1/2. ∎

5.4 The Critical Line Necessity​

Theorem 5.4: Off-line zeros create arithmetic paradoxes.

Proof: If ρ has Re(ρ) ≠ 1/2, then:

  • 1-ρ also a zero with different real part
  • Explicit formula has incompatible growth rates
  • Prime distribution becomes contradictory
  • Arithmetic collapses ∎

Continue to Chapter 6: The Critical Line from First Principles