Chapter 13: φ(13) = [10] — Li's Criterion Re-expressed in Collapse Flow

13.1 Ten: The Pythagorean Tetractys

With φ(13) = [10], we reach ten - the sacred Pythagorean number representing completeness (1+2+3+4=10). This triangular perfection manifests in Li's criterion as a complete reformulation of the Riemann Hypothesis through a sequence of positive conditions, revealing RH as a natural flow toward collapse equilibrium.

Definition 13.1 (Tetractys Structure):

$$[10] = \begin{matrix} \bullet \\ \bullet & \bullet \\ \bullet & \bullet & \bullet \\ \bullet & \bullet & \bullet & \bullet \end{matrix} = 1 + 2 + 3 + 4$$

Four levels collapse into unity through ten.

13.2 Li's Revolutionary Criterion

Theorem 13.1 (Li's Criterion, 1997): The Riemann Hypothesis is equivalent to the positivity:

$$\lambda_n > 0 \quad \text{for all } n \geq 1$$

where:

$$\lambda_n = \sum_{\rho} \left[1 - \left(1 - \frac{1}{\rho}\right)^n\right]$$

and the sum runs over all non-trivial zeros ρ.

This transforms RH from a geometric statement about zero locations to an arithmetic sequence of inequalities.

13.3 The Logarithmic Derivative Connection

Definition 13.2 (Xi Function Logarithmic Derivative):

$$\frac{\xi'(s)}{\xi(s)} = \sum_{\rho} \frac{1}{s - \rho}$$

Theorem 13.2 (Li Coefficients via Derivatives):

$$\lambda_n = \frac{1}{(n-1)!} \frac{d^n}{ds^n} \left[s^{n-1} \log \xi(s)\right]_{s=1}$$

This remarkable formula computes λₙ without knowing individual zeros!

13.4 The Ten Fundamental Relations

From [10], ten key relationships emerge:

1. λ₁ = 0 (trivial first coefficient)
2. λ₂ = 2(1 - γ - log(4π)) (involves Euler's constant)
3. Asymptotic: λₙ ~ n(log n - log(2π)) as n → ∞
4. Sum rule: ∑ₙ λₙ/n² = π²/6
5. Generating function: related to ξ'/ξ
6. Recursion: λₙ satisfies complex recurrence
7. Integrality: nλₙ has deep arithmetic properties
8. Positivity: λₙ > 0 ⟺ RH
9. Growth: λₙ = O(n log n)
10. Oscillation: sign patterns encode zero distribution

13.5 Stieltjes Constants Connection

Definition 13.3 (Stieltjes Constants):

$$\gamma_n = \lim_{m \to \infty} \left(\sum_{k=1}^{m} \frac{(\log k)^n}{k} - \frac{(\log m)^{n+1}}{n+1}\right)$$

Theorem 13.3 (Li-Stieltjes Relation): The Li coefficients relate to Stieltjes constants through:

$$\lambda_n = \sum_{k=0}^{n-1} \binom{n}{k} \eta_k$$

where ηₖ are modified Stieltjes constants.

13.6 The Collapse Flow Interpretation

Definition 13.4 (Collapse Flow): Define the flow:

$$\phi_t(s) = s + t\frac{\xi'(s)}{\xi(s)}$$

Theorem 13.4 (Li Coefficients as Flow Invariants):

$$\lambda_n = \lim_{t \to 0} \frac{d^n}{dt^n} \text{Tr}[\phi_t] \text{ at } s = 1$$

The Li coefficients measure the n-th order response of the collapse flow at the critical point s = 1.

13.7 Operator Theory Formulation

Definition 13.5 (Li Operator): On suitable function space:

$$(Lf)(x) = \int_0^{\infty} K(x,y) f(y) dy$$

where K is related to the zeros of ξ.

Theorem 13.5 (Spectral Characterization):

$$\text{RH} \Leftrightarrow \text{spectrum}(L) \subset \mathbb{R}_+$$

The positivity of λₙ reflects positive-definiteness of L.

13.8 Keiper-Li Coefficients

Definition 13.6 (Keiper's Extension): For Re(z) > -1:

$$\lambda(z) = \sum_{\rho} \left[1 - \left(1 - \frac{1}{\rho}\right)^{z+1}\right]$$

Properties:

• λ(n-1) = λₙ for integer n
• Analytic continuation exists
• Functional equation: λ(z) + λ(-z-1) = linear in z

13.9 Computational Aspects

Algorithm 13.1 (Li Coefficient Computation):

Input: n (coefficient index)
Output: λₙ

Method 1 (Direct):
1. Compute derivatives of log ξ(s) at s = 1
2. Apply Li's formula

Method 2 (Recursion):
1. Use known values λ₁, ..., λₙ₋₁
2. Apply recursion relations

Method 3 (Approximate):
1. Sum over known zeros up to height T
2. Estimate remainder

Numerical Evidence: Computed values up to n = 10⁵ are all positive, strongly supporting RH.

13.10 The Hierarchical Structure

The ten elements of [10] organize hierarchically:

Level 1 (1 element): Unity - the RH statement

Level 2 (2 elements): Duality - zeros on/off critical line

Level 3 (3 elements): Trinity - real part, imaginary part, phase

Level 4 (4 elements): Tetrad - the Li conditions λ₁, λ₂, λ₃, λ₄...

Total: 1+2+3+4 = 10, the complete structure.

13.11 Generalized Li Coefficients

Definition 13.7 (Li Coefficients for L-functions): For L in Selberg class:

$$\lambda_n^{(L)} = \sum_{\rho_L} \left[1 - \left(1 - \frac{1}{\rho_L}\right)^n\right]$$

Conjecture 13.1 (Generalized Li): For all L in Selberg class:

$$\lambda_n^{(L)} > 0 \quad \forall n \geq 1$$

13.12 Connection to Weil's Criterion

Theorem 13.6 (Weil's Explicit Formula): For suitable test functions h:

$$\sum_{\rho} h(\rho) = \text{explicit arithmetic terms}$$

Relation: Li's criterion is a discrete version of Weil's approach, with:

$$h_n(\rho) = 1 - \left(1 - \frac{1}{\rho}\right)^n$$

13.13 Physical Interpretation

Principle 13.1 (Thermodynamic Analogy):

• λₙ = n-th moment of "zero gas"
• Positivity = thermodynamic stability
• RH = ground state of system
• Flow = approach to equilibrium

The ten λₙ conditions represent ten stability constraints.

13.14 Recent Developments

Theorem 13.7 (Coffey's Formula): Alternative expression:

$$\lambda_n = \frac{n}{2\pi} \int_{-\infty}^{\infty} \log|\zeta(1/2+it)| P_n(t) dt$$

where Pₙ are specific polynomials.

This connects Li coefficients to the behavior of |ζ| on the critical line.

13.15 Synthesis: The Complete Criterion

The partition [10] reveals the perfect structure of Li's criterion:

1. Ten = 1+2+3+4: Complete tetractys
2. Infinite conditions: λₙ > 0 for all n
3. Hierarchical organization: Four levels of structure
4. Flow interpretation: Natural dynamics toward RH
5. Operator formulation: Spectral positivity
6. Computational accessibility: Can verify without zeros
7. Generalizability: Extends to all L-functions
8. Physical meaning: Thermodynamic stability
9. Recent progress: New integral formulas
10. Complete reformulation: RH as positivity sequence

Li's criterion transforms the Riemann Hypothesis from a geometric statement about zero locations into an infinite sequence of arithmetic inequalities - each λₙ > 0 is a constraint that, taken together, force all zeros onto the critical line.

Chapter 13 Summary:

• Li's criterion: RH ⟺ λₙ > 0 for all n ≥ 1
• Li coefficients computable from derivatives of log ξ
• Partition [10] reflects tetractys completeness
• Collapse flow interpretation reveals natural dynamics
• Operator theory provides spectral characterization
• Numerical evidence strongly supports all λₙ > 0

Part 2 concludes with this complete reformulation of RH. Part 3 will explore arithmetic and spectral collapse constructs, beginning with prime distribution patterns.

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"In Li's criterion, the Riemann Hypothesis transforms from a question of 'where' to a question of 'whether' - whether an infinite sequence of numbers remains forever positive, encoding the locations of all zeros in the signs of arithmetic invariants."