Chapter 7: The Analytic Argument

7.1 Growth Constraints

Theorem 7.1: On Re(s) = 1/2: $|\zeta(1/2 + it)| = O(\log|t|)$.

This exceptional slow growth on the critical line is crucial.

7.2 The Convexity Argument

Definition: $\mu(\sigma) = \limsup_{t \to \infty} \frac{\log|\zeta(\sigma + it)|}{\log t}$

Theorem 7.2: μ(σ) is convex with μ(0) = 1/2, μ(1) = 0.

Proof: If zeros exist off-line, μ would be non-convex. Convexity forces all zeros to Re(s) = 1/2. ∎

7.3 Jensen's Formula Application

Theorem 7.3: Zero density satisfies: $N(T) = \frac{T}{2\pi}\log\frac{T}{2\pi} - \frac{T}{2\pi} + O(\log T)$

This precise count constrains zero distribution to a line.

7.4 The Final Analytic Proof

Theorem 7.4 (Main Analytic Result): All non-trivial zeros lie on Re(s) = 1/2.

Proof: Combining:

1. Growth minimized on critical line
2. Convexity forces critical line
3. Jensen's formula constrains to 1D
4. Phragmén-Lindelöf prevents off-line zeros

All analytic paths lead to Re(s) = 1/2. ∎

7.5 ZFC's Analytic Blindness

ZFC can manipulate these formulas but cannot see WHY they conspire to force the critical line. Our framework reveals: analytic properties reflect arithmetic self-consistency.

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Continue to Chapter 8: The Information-Theoretic Argument