Mertens Function
number_theory · 36 views
number_theory
What It Is
Cumulative sum of the Moebius function --- a random-walk-like sequence whose growth rate is equivalent to the Riemann Hypothesis (O(n^(½+ε)) iff RH)
Interpretation
Standard analysis sees: left-skewed; smooth / autocorrelated; long-range memory (persistent); strongly periodic; volatility-clustering (bursty); nonstationary / drifting. The atlas finds no named structure, but the source is distinctively extreme on H3 Icosahedral:temporal_coherence (+3.3z) — beyond what the standard bank predicts for it.
What standard analysis sees
tail heaviness0.49
asymmetry0.05
occupancy0.54
short-range corr0.92
long-range memory0.91
spectral colour0.26
periodicity0.95
complexity0.41
time-irreversibility0.66
volatility clustering0.92
multifractality0.39
dimensionality0.17
nonstationarity0.91
What the atlas adds
Atlas-extreme metrics the standard bank can’t predict for this source
H3 Icosahedral:temporal_coherence | +3.3z | bank-miss 2.2σ |
H3 Icosahedral:nn_enrichment | +3.0z | bank-miss 2.1σ |
Möbius-S³:phi_return_cv | +2.8z | bank-miss 1.6σ |
Chladni:nodal_clustering | +2.4z | bank-miss 1.4σ |
Composition
dtypefloat64
range[-132, 68]
unique values201 / 16384
mean ± std-28.9 ± 53.7
Render Gallery
Native > phase_plotraw
Native > timeseriesraw
Native > triple_latticeraw
Boltzmann > lag_map
Boltzmann > spin_grid
Cayley > default
Chladni > plate_bottom
Chladni > plate_top
_(centered)/signed_log_z/Mertens_Function.png)
Heisenberg (Nil) (centered) > signed_log_z
_(centered)/xy_path/Mertens_Function.png)
Heisenberg (Nil) (centered) > xy_path
Hodge–Laplacian > default
/barcode/Mertens_Function.png)
Inflation (Substitution) > barcode
/d_curve/Mertens_Function.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/Mertens_Function.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/Mertens_Function.png)
Ulam Spiral (Sacks) > default
/default/Mertens_Function.png)
Ulam Spiral (Square) > default
Visibility Graph > default
Wavelet Cascade > default
Atlas Position
| Nearest neighbor | Distance | |
|---|---|---|
| Rudin-Shapiro | 3.02 | |
| Hilbert Walk | 3.09 | cross-domain |
| Brownian Walk | 3.19 | cross-domain |
Which Geometries Light Up
Chladni › Chladni:nodal_clustering | rank 4/298 | 7.8509 |
H3 Icosahedral › H3 Icosahedral:temporal_coherence | rank 4/298 | 0.3147 |
Inflation (Substitution) › Inflation (Substitution):return_concentration | rank 296/298 | 0.0806 |
Sol (Thurston) › Sol (Thurston):sol_step_persistence | rank 3/298 | 0.8975 |
Zariski › Zariski:residual_slope | rank 294/298 | -4.7724 |
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