Riemann Hypothesis
Numerical Toolkit

§1. The Riemann Hypothesis

The Riemann Hypothesis (1859) states that all non-trivial zeros of the Riemann zeta function ζ(s) have real part ½. First proposed by Bernhard Riemann in his 1859 memoir, it is one of Hilbert's 23 problems (1900) and a Clay Millennium Prize problem ($1,000,000).

The function extends meromorphically to all of ℂ via the functional equation ζ(s) = 2^sπ^s−1 sin(πs/2) Γ(1−s) ζ(1−s). Trivial zeros occur at s = −2, −4, −6, …. Non-trivial zeros lie in the critical strip 0 < Re(s) < 1. The RH asserts they all lie on the critical line Re(s) = ½.

Current Status: Over 10¹³ zeros verified on the critical line (Platt & Trudgian, 2021). No counterexample found. The $1,000,000 Clay prize remains unclaimed.

§2. The Riemann–Siegel Formula

On the critical line, the Hardy Z-function Z(t) = e^iθ(t)ζ(½+it) is real-valued. Its zeros correspond to ζ zeros. The Riemann–Siegel formula computes Z(t) efficiently:

where N = ⌊√(t/(2π))⌋ and R(t) is the Gabcke remainder series with coefficients C_k(u). Our implementation uses M = 10 remainder terms, providing ~10⁻¹⁴ accuracy for t > 10.

from rh_numerical.zeta import riemann_siegel_Z
r = riemann_siegel_Z(100.0, M=10)
# r.Z = Z(100), r.theta = θ(100), r.N = 3 main-sum terms

§3. Zero Finding & Verification

Zero Location: Evaluate Z(t) on a grid (spacing δ = 0.1), detect sign changes, refine via bisection. Gram Points g_n satisfy θ(g_n) = nπ and typically bracket one zero each. Turing's Method (1953): Verifies no zeros are missed by tracking discrepancy D(t) = N_expected(t) − N_found(t) against the Lehman bound |∫S(t)dt| ≤ 1.91 + 0.114 log(t/2π).

from rh_numerical.zeros import find_zeros_up_to, N_T_expected
zeros = find_zeros_up_to(100.0, step=0.1, M=10)
# Found 29 zeros; N(100) ≈ 29.00; discrepancy = 0.00

Riemann–von Mangoldt Formula: N(T) = (T/2π)log(T/2π) − T/2π + 7/8 + S(T) + O(1/T). Zeros become denser with height: mean density ~ (1/2π)log(T/2π).

§4. Equivalent Formulations

RH admits 100+ equivalent forms. Four computationally checkable criteria:

from rh_numerical.equivalences import robin_check, li_criterion_check
holds, sigma, bound = robin_check(10080)  # σ=39312, ratio=0.9858 ✓
li = li_criterion_check(10)                # All λ_n > 0 ✓

§5. GUE & Random Matrix Theory

Montgomery–Dyson (1972): Pair correlation of Riemann zeros matches GUE eigenvalue statistics: R₂(x) = 1 − sinc²(πx). Wigner surmise: P(s) = (32/π²)s²e^−4s2/π for GUE (β=2). Zeros exhibit quadratic level repulsion P(s) ∝ s² as s → 0, in contrast to Poisson (no repulsion) or GOE (linear, β=1). Spectral rigidity: Number variance Σ²(L) ~ (1/π²)log L, far more rigid than Poisson Σ²(L) = L.

§6. Statistical Modelling

Ensemble Simulators: generate_gue(N), generate_goe(N), generate_gse(N) produce eigenvalue distributions from the three classical Gaussian ensembles. Tracy–Widom: CDF/PDF for β=1,2,4 describing the fluctuations of the largest eigenvalue. Dyson β Estimation: Log-log regression on small spacings estimates the repulsion exponent; bootstrap provides 95% confidence intervals. Bayesian RH Test: Computes Bayes factor comparing H₀ (β=2, RH true) vs. H₁ (β≠2, RH false).

from rh_advanced.ensembles import generate_gue, estimate_dyson_beta, ks_test_gue
gue = generate_gue(500, seed=42)
beta = estimate_dyson_beta(spacings)  # {beta: 2.03, ci_95: (1.92, 2.14)}
ks = ks_test_gue(spacings)           # {D_statistic: 0.04, reject_gue: False}

§7. Machine Learning Layer

SpacingVAE: Variational Autoencoder trained on zero spacings generates synthetic spacing distributions statistically indistinguishable from GUE. NeuralBetaEstimator: Feed-forward network predicts β directly from spacing histograms, trained on synthetic ensemble data. GaussianProcessZeta: GP regression with RBF kernel interpolates Z(t) with uncertainty estimates—useful for adaptive sampling and anomaly detection.

from rh_advanced.ml_layer import SpacingVAE, NeuralBetaEstimator
vae = SpacingVAE(latent_dim=4, hidden_dim=32)
vae.fit(spacing_windows, epochs=300)
synthetic = vae.generate(1000)  # statistically ~ GUE

nn = NeuralBetaEstimator(); nn.fit(n_train=3000)
beta_nn = nn.predict(spacings)  # neural β estimate

§8. High-Performance Computing

Parallel Z(t): parallel_evaluate_Z() distributes across CPU cores via ProcessPoolExecutor. Adaptive Zero Finding: Step-size adapts to Z(t) magnitude—fine near extrema, coarse near zeros. MemoizedZeta: LRU cache with 100K entries provides 90%+ hit rates for repeated evaluations. Benchmark: C++ achieves 10–100× speedup over Python for large-scale zero verification.

§9. Connes Spectral Operator

The Connes–Consani–Moscovici (2025) zeta spectral triple constructs a Galerkin matrix whose eigenvalues converge to Riemann zeros. Our implementation computes the finite-dimensional approximation and compares predicted zeros with known values. LMFDB Integration: Access to 103+ billion Platt-verified zeros. Dirichlet L-functions: L(s,χ) computation for GRH exploration.

from rh_advanced.connes_lmfdb import connes_galerkin_matrix, dirichlet_L
result = connes_galerkin_matrix(cutoff=19.0, N_max=60)
# Mean relative error vs actual zeros: ~0.05

§10. REST API Reference

Production FastAPI server with 26 endpoints at http://localhost:8420.

python -m rh_services.api
curl "http://localhost:8420/zeta/t/100.0"
curl "http://localhost:8420/advanced/number-model?t_max=300"

§11. C++ Library

Header-optimized C++17 library in cpp/. CMake build with OpenMP, AVX2/FMA. The Riemann–Siegel main sum uses OpenMP #pragma omp parallel for. Gabcke coefficients via Haselgrove polynomial evaluation. Performance: Z(100) in 0.3 μs (30× Python), Z(10⁴) in 30 μs (100× Python).

cd cpp && mkdir build && cd build
cmake .. -DCMAKE_BUILD_TYPE=Release -DRIEMANN_USE_OPENMP=ON
make -j$(nproc)
./test_riemann && ./bench_riemann

§12. Colab Notebooks

Seven interactive notebooks for browser-based exploration:

Open any notebook: github.com/twomathematicians-code/riemann-hypothesis/tree/master/notebooks

§13. Installation & Quick Start

git clone https://github.com/twomathematicians-code/riemann-hypothesis.git
cd riemann-hypothesis
pip install -r rh_services/requirements.txt
python -m rh_numerical.main --quick
python -m rh_services.api          # → http://localhost:8420

Dependencies: Python 3.10+, NumPy, SciPy, FastAPI, uvicorn, Pydantic. Optional: mpmath (high-precision), matplotlib (plots), C++17 compiler + CMake (C++ library). Docker: docker build -t rh-services -f rh_services/Dockerfile . && docker run -p 8420:8420 rh-services

§14. Repository Structure

riemann-hypothesis/
├── rh_numerical/          Python numerical toolkit (6 modules)
│   ├── zeta.py            Riemann–Siegel Z(t), θ(t), Euler product
│   ├── zeros.py           Zero finding, Gram points, Gram's law
│   ├── turing.py          Turing verification, S(T), Lehman bounds
│   ├── equivalences.py    Robin, Lagarias, Li, Mertens
│   ├── correlations.py    Pair correlation, GUE stats, Wigner
│   └── visualization.py   10+ plot types + dashboard
├── rh_advanced/           Advanced computing framework (7 modules)
│   ├── ensembles.py       GUE/GOE/GSE, Tracy–Widom, Dyson β
│   ├── predictive.py      Bayesian RH test, anomaly detection
│   ├── ml_layer.py        VAE, Neural β, Gaussian Process
│   ├── dyson_brownian.py  Dyson BM, Log-gas MC, TW MLE
│   ├── connes_lmfdb.py    Connes operator, LMFDB, Dirichlet L
│   ├── hpc.py             Parallel, adaptive, memoized
│   └── number_model.py    5-layer NumberModel synthesis
├── rh_services/           REST API service layer (8 files)
├── cpp/                   C++17 high-performance library
├── notebooks/             7 Colab/Kaggle notebooks
└── docs/                  GitHub Pages + Handbook

§15. Verified Results