Chapter 14: φ(14) = [10,2] — RH as Prime Distribution Collapse: From ψ(x) to π(x)

14.1 The Partition [10,2]: Completeness Plus Duality

With φ(14) = [10,2], we see the tetractys (10) accompanied by fundamental duality (2). This structure perfectly captures how the Riemann Hypothesis controls the transition from the smooth Chebyshev function ψ(x) to the discrete prime counting function π(x), revealing RH as the condition for optimal error terms.

Definition 14.1 (Dual Completeness):

$$[10,2] = \lbrace\lbrace 1,...,10\rbrace, \lbrace 11,12\rbrace\rbrace$$

Complete structure (10) plus binary refinement (2).

14.2 The Prime Counting Functions

Definition 14.2 (Fundamental Functions):

$$\begin{align} \pi(x) &= \#\lbrace p \leq x : p \text{ prime}\rbrace \\ \theta(x) &= \sum_{p \leq x} \log p \\ \psi(x) &= \sum_{p^k \leq x} \log p = \sum_{n \leq x} \Lambda(n) \end{align}$$

Theorem 14.1 (Relationships):

$$\psi(x) = \theta(x) + O(\sqrt{x})$$ $$\pi(x) = \frac{\theta(x)}{\log x} + \int_2^x \frac{\theta(t)}{t(\log t)^2} dt$$

14.3 The Explicit Formula

Theorem 14.2 (Riemann-von Mangoldt): For x not a prime power:

$$\psi(x) = x - \sum_{\rho} \frac{x^\rho}{\rho} - \log(2\pi) - \frac{1}{2}\log\left(1-\frac{1}{x^2}\right)$$

where the sum runs over all non-trivial zeros ρ of ζ(s).

This remarkable formula expresses prime distribution through zeta zeros!

14.4 The Ten Components of Prime Distribution

From [10], ten aspects emerge:

1. Main term: x (linear growth)
2. Zero contribution: -∑ x^ρ/ρ (oscillations)
3. Constant: -log(2π) (normalization)
4. Tail correction: -½log(1-1/x²)
5. Trivial zeros: Implicit in formula
6. Phase information: arg(ρ) creates interference
7. Amplitude control: 1/|ρ| weights
8. Density effects: More zeros = finer oscillations
9. Error bounds: Depend on zero locations
10. Completeness: All arithmetic encoded

14.5 The Duality [2]: Smooth vs Discrete

The [2] component represents the fundamental duality:

Smooth Function: ψ(x) - continuous with jumps at prime powers

Discrete Function: π(x) - pure step function at primes

Connection:

$$\pi(x) = \sum_{n \leq x} \frac{\mu(n)}{n} \psi(x^{1/n})$$

where μ is the Möbius function.

14.6 Prime Number Theorem and RH

Theorem 14.3 (Prime Number Theorem):

$$\pi(x) \sim \frac{x}{\log x}$$

Theorem 14.4 (Error Terms):

• Unconditional: π(x) = Li(x) + O(xe^(-c√log x))
• Under RH: π(x) = Li(x) + O(√x log x)

where Li(x) = ∫₂ˣ dt/log t is the logarithmic integral.

14.7 The Collapse Interpretation

Definition 14.3 (Prime Collapse): The transition ψ(x) → π(x) represents:

$$\text{Weighted count} \xrightarrow{\text{collapse}} \text{Pure count}$$

Theorem 14.5 (Optimal Collapse under RH): RH ensures the oscillations from zeros in ψ(x) optimally cancel when extracting π(x).

14.8 Von Koch's Result

Theorem 14.6 (Von Koch, 1901): The Riemann Hypothesis is equivalent to:

$$\pi(x) = \text{Li}(x) + O(\sqrt{x} \log x)$$

RH provides the best possible error term in the Prime Number Theorem.

14.9 Oscillation Analysis

Definition 14.4 (Normalized Error):

$$E(x) = \frac{\pi(x) - \text{Li}(x)}{\sqrt{x}/\log x}$$

Theorem 14.7 (Littlewood): E(x) oscillates between +∞ and -∞, changing sign infinitely often.

The zeros create these oscillations through:

$$\sum_{\rho} \frac{x^\rho}{\rho} = \sum_{\rho} \frac{x^{1/2+i\gamma}}{\rho}$$

14.10 The Cramér Model

Probabilistic Model: Cramér proposed primes behave like random integers with probability 1/log n at n.

Prediction: Under this model:

$$\pi(x) - \text{Li}(x) = O(\sqrt{x} \log \log x)$$

But RH only gives O(√x log x) - the log x factor remains mysterious.

14.11 Skewes' Number

Definition 14.5 (First Crossover): The smallest x where π(x) > Li(x).

Bounds:

• Skewes (1933): x < 10^10^10^34 (assuming RH)
• Current: x < e^727.95 ≈ 10^316

This enormous number shows how long Li(x) dominates despite eventual oscillation.

14.12 The Twelve Aspects Combined

The 14 = 10 + 2 + 2 structure suggests three information levels:

Ten Core Components: As listed in 14.4

First Duality: Smooth ψ(x) vs discrete π(x)

Second Duality: Overestimate vs underestimate in Li(x) - π(x)

14.13 Explicit Bounds

Theorem 14.8 (Schoenfeld): Assuming RH, for x ≥ 2657:

$$|\pi(x) - \text{Li}(x)| < \frac{\sqrt{x} \log x}{8\pi}$$

Unconditional (Dusart): For x ≥ 467746:

$$\frac{x}{\log x - 1} < \pi(x) < \frac{x}{\log x - 1.1}$$

14.14 Computational Verification

Algorithm 14.1 (Prime Count Verification):

Input: x (upper bound)
Output: Verification of error bounds

1. Compute π(x) by sieving
2. Compute Li(x) by numerical integration
3. Compute ψ(x) using prime factorization
4. Verify explicit formula using known zeros
5. Check error term growth
6. Test oscillation changes

Results: Computations up to 10²⁷ confirm RH predictions.

14.15 Synthesis: Distribution Collapse

The partition [10,2] illuminates the prime distribution collapse:

1. Ten components in explicit formula create complete description
2. First duality: Smooth ψ(x) collapses to discrete π(x)
3. Second duality: Li(x) alternately over/underestimates
4. RH ensures optimal O(√x log x) error
5. Zeros orchestrate the oscillations
6. Phase cancellation from ρ and 1-ρ pairing
7. Amplitude decay as 1/|ρ| ~ 1/γ
8. Littlewood oscillation guarantees sign changes
9. Skewes' bound shows initial bias
10. Computational evidence supports RH control
11. Cramér conjecture suggests even better might be true
12. Complete collapse from analysis to arithmetic

The Riemann Hypothesis emerges as the optimal collapse principle - ensuring that the continuous analytical information in ζ(s) collapses into discrete prime counts with minimal error.

Chapter 14 Summary:

• Prime distribution encoded in zeros via explicit formula
• ψ(x) = x - ∑x^ρ/ρ + small terms
• RH ⟺ π(x) = Li(x) + O(√x log x)
• Partition [10,2] reflects completeness plus duality
• Zeros create oscillations that optimally cancel
• Computational evidence strongly supports RH control

Chapter 15 explores φ(15) = [11], where the Hilbert-Pólya conjecture suggests zeros arise from a self-adjoint operator's spectrum.

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"In the dance between π(x) and Li(x), we see the Riemann Hypothesis at work - conducting an infinite orchestra of zeros to ensure that chaos resolves into the tightest possible order, that the wildness of primes submits to analytic law."