M8n Pattern Analysis + ACE  ·  Live Computation

Riemann Hypothesis:
Zeta Zero Entropy Analysis

The Riemann Hypothesis — unsolved for 165 years — asks whether all non-trivial zeros of ζ(s) lie on the critical line Re(s) = ½. This demo uses M8n Pattern Analysis to compute zeros with mpmath, classify them into Gram intervals, measure conditional entropy collapse, and fit a Markov transfer operator — methods that quantify the hidden deterministic structure predicted by the Hilbert–Pólya conjecture.

ζ(s) = Σ n⁻ˢ · Zeros at s = ½ + it · H₁ = −Σ p(m) log p(m)
Ready
m8n:pattern-analysis:rh — mpmath + ACE
ζ

Press Run live analysis to compute non-trivial Riemann zeta zeros, classify them into Gram intervals, and measure entropy collapse in real time.

Uses mpmath.zetazero() · Gram interval classification · Shanker (2013) entropy method

📊 Live Statistics
Zeros computed
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Gram intervals
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H₁ (nats)
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Spectral gap
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📶 Zero Distribution by Gram Interval
Run analysis to see distribution
📉 Conditional Entropy Collapse Fₙ
Run analysis to see entropy collapse
🔀 Markov Transition Matrix
Run analysis to see transition matrix
🧠 ACE Transfer Operator Eigenvalues
Run analysis to see eigenvalue spectrum
📚 Literature Comparison (Shanker 2013 · N=10⁶ zeros)
p(0) — zero zeros in interval
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0.170 (lit.)
p(1) — one zero in interval
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0.661 (lit.)
p(2) — two zeros in interval
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0.166 (lit.)
H₁ (Shannon entropy)
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~0.88 nats (lit.)
F₁ / H₁ (normalised cond. entropy)
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~0.62 at N=1 → 0 (lit.)
⚛️
Method: Shanker, O. (2013). "Good to one-side Gram's law for the Riemann zeros." Ramanujan J. 31, 273–284. Gram interval classification based on the Riemann-Siegel theta function θ(t). Zeros computed via mpmath.zetazero() (F. Johansson, 2023). Markov eigenvalue analysis: NumPy linear algebra.

M8n functions used: Pattern Analysis · ACE (Adaptive Cognitive Engine) — same API available via the Vending Machine for any sequence dataset.
Apply this analysis to your data → API documentation