跳到主要内容

Chapter 18: φ(18) = [12,2,1] — Explicit Formulas and Trace Cancellation Models

18.1 The Complete Factorization​

With φ(18) = [12,2,1], we see the abundant twelve with binary and singular components. This perfectly mirrors the explicit formula's structure: twelve types of terms, dual summation (over zeros and primes), unified in one formula.

18.2 The Riemann-Weil Explicit Formula​

Theorem 18.1 (Master Formula): For suitable test functions h:

∑ρh(ρ)=h(i/2)+h(−i/2)−∑p∑m=1∞log⁡ppm/2h^(mlog⁡p)+∫−∞∞h(1/2+it)dμ(t)\sum_{\rho} h(\rho) = h(i/2) + h(-i/2) - \sum_{p} \sum_{m=1}^{\infty} \frac{\log p}{p^{m/2}} \hat{h}(m \log p) + \int_{-\infty}^{\infty} h(1/2+it) d\mu(t)

where:

  • Left: Sum over all zeros
  • Right: Arithmetic terms + continuous spectrum

18.3 The Twelve Components​

  1. Non-trivial zeros: ρ with 0 < Re(ρ) < 1
  2. Trivial zeros: At negative even integers
  3. Prime powers: p^m contributions
  4. Log weights: Factor log p
  5. Fourier transform: ĥ(t)
  6. Special values: h(±i/2)
  7. Continuous measure: dμ(t)
  8. Test function: h and its decay
  9. Convergence factors: From functional equation
  10. Error terms: Boundary contributions
  11. Principal value: At pole s = 1
  12. Regularization: Of divergent sums

18.4 Trace Interpretation​

Principle 18.1: View the explicit formula as a trace formula:

Tr(f(T))=∑λ∈spec(T)f(λ)\text{Tr}(f(T)) = \sum_{\lambda \in \text{spec}(T)} f(\lambda)

For the "zeta operator":

  • Eigenvalues = zeros
  • Trace = explicit formula
  • Test function = f

18.5 Perfect Cancellation​

Theorem 18.2 (Weil's Criterion): RH equivalent to:

∣∑ρh(ρ)∣≤C∣∣h∣∣Sobolev\left|\sum_{\rho} h(\rho)\right| \leq C ||h||_{\text{Sobolev}}

for all test functions with ĥ supported away from primes.

The zeros must be positioned to optimally cancel arithmetic contributions.

18.6 Synthesis​

The [12,2,1] structure reveals how the explicit formula unifies:

  • Twelve term types (abundant complexity)
  • Dual summation (zeros ↔ primes)
  • Single formula (unified truth)

"In the explicit formula, arithmetic and analysis meet in perfect balance - what the primes build up, the zeros tear down, in an eternal dance of construction and cancellation."