Riemann-Hardy-Littlewood
number-theory · 37 views
number-theoryarithmetic
What It Is
Riemann's 1859 'monster' R(t) = sum_{n>=1} sin(pi n^2 t)/(pi n^2). Continuous everywhere, differentiable only at rationals (2p+1)/(2q+1) where R' = -1/2 (Gerver 1970), nowhere else (Hardy 1916). Quadratic frequencies n^2 with 1/n^2 amplitude decay --- the steepest known cosine fractal: ~92% of energy in the fundamental, non-differentiability emerging from the long thin Hausdorff-dimensional tail.
Interpretation
Standard analysis sees: smooth / autocorrelated; long-range memory (persistent); strongly periodic; volatility-clustering (bursty); low-dimensional; nonstationary / drifting. The atlas additionally detects Multi-Scale Wasserstein:w_max_ratio, Nonstationarity:adf_pvalue, Multifractal Spectrum:hurst_estimate.
What standard analysis sees
tail heaviness0.17
asymmetry0.40
occupancy0.82
short-range corr0.99
long-range memory0.99
spectral colour0.30
periodicity0.99
complexity0.47
time-irreversibility0.85
volatility clustering0.97
multifractality0.57
dimensionality0.11
nonstationarity0.90
What the atlas adds
Multi-Scale Wasserstein:w_max_ratio+8.1z
scale-coherent dispersion mismatch (max/min Wasserstein across scales)
Nonstationarity:adf_pvalue+3.8z
unit-root nonstationarity (ADF cannot reject random-walk null)
Multifractal Spectrum:hurst_estimate+2.0z
self-affinity H = τ(2)/2 (alternate to Hölder hurst)
Atlas-extreme metrics the standard bank can’t predict for this source
Septagonal (Danzer):z_primary | +3.2z | bank-miss 1.1σ |
Composition
dtypefloat64
range[-0.4, 0.4]
unique values16384 / 16384
mean ± std-1.81e-05 ± 0.234
Render Gallery
Native > phase_plotraw
Native > timeseriesraw
Native > triple_latticeraw
Boltzmann > lag_map
Boltzmann > spin_grid
Cayley > default
Chladni > free_plate_nodal
Chladni > plate_bottom
Chladni > plate_top
_(centered)/signed_log_z/Riemann-Hardy-Littlewood.png)
Heisenberg (Nil) (centered) > signed_log_z
_(centered)/xy_path/Riemann-Hardy-Littlewood.png)
Heisenberg (Nil) (centered) > xy_path
Hodge–Laplacian > default
/barcode/Riemann-Hardy-Littlewood.png)
Inflation (Substitution) > barcode
/d_curve/Riemann-Hardy-Littlewood.png)
Inflation (Substitution) > d_curve
Information Theory > default
Julia Set > escape_histogram
Klein Bottle > default
Laplacian > default
Logarithmic Spiral > default
Möbius-S³ > mobius_cylinder
Möbius-S³ > s2_base_map
Ordinal Partition > default
/phi_spectrum/Riemann-Hardy-Littlewood.png)
Penrose (Quasicrystal) > phi_spectrum
Persistent Homology > barcode
Persistent Homology > h1_diagram
Predictability > default
Recurrence Quantification > default
Spectral Analysis > default
Spectral Graph > default
Spirograph > mode_spectrum
Spirograph > phasor_trajectory
Symplectic > flux_grid
Symplectic > phase_space
/default/Riemann-Hardy-Littlewood.png)
Ulam Spiral (Sacks) > default
/default/Riemann-Hardy-Littlewood.png)
Ulam Spiral (Square) > default
Visibility Graph > default
Wavelet Cascade > default
Atlas Position
| Nearest neighbor | Distance | |
|---|---|---|
| fBm (Persistent) | 3.64 | cross-origin |
| Perlin Noise | 4.00 | cross-origin |
| Geometric Brownian Motion | 4.03 | cross-origin |
Which Geometries Light Up
Ammann-Beenker (Octagonal) › Ammann-Beenker (Octagonal):convergent_profile | rank 305/307 | -0.1570 |
Catch24 › Catch24:CO_HistogramAMI_even_2_5 | rank 1/307 | 1.5593 |
Catch24 › Catch24:SC_FluctAnal_2_dfa_50_1_2_logi_prop_r1 | rank 1/307 | 0.8720 |
Catch24 › Catch24:SP_Summaries_welch_rect_area_5_1 | rank 3/307 | 0.9999 |
Catch24 › Catch24:FC_LocalSimple_mean3_stderr | rank 305/307 | 0.0089 |
Chladni › Chladni:plate_low_mode_fraction | rank 3/307 | 0.9955 |
H² × ℝ (Thurston) › H² × ℝ (Thurston):radial_temporal_memory | rank 4/307 | 4.9648 |
Multi-Scale Wasserstein › Multi-Scale Wasserstein:w_max_ratio | rank 2/307 | 2884.9213 |
Navier-Stokes › Navier-Stokes:sl_fit_quality | rank 1/307 | 0.9579 |
Navier-Stokes › Navier-Stokes:ess_rescaling_gain | rank 303/307 | 0.0001 |
Predictability › Predictability:excess_predictability | rank 2/307 | 2.9204 |
SL(2,ℝ) (Thurston) › SL(2,ℝ) (Thurston):trace_autocorrelation | rank 4/307 | 0.9992 |
Septagonal (Danzer) › Septagonal (Danzer):cubic_coherence | rank 3/307 | 0.2564 |
Septagonal (Danzer) › Septagonal (Danzer):z_primary | rank 3/307 | 0.3143 |
Spectral Analysis › Spectral Analysis:spectral_bandwidth | rank 304/307 | 0.0009 |
Spherical S² › Spherical S²:hemisphere_balance | rank 3/307 | 0.9999 |
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