Mertens Function
number-theory · 37 views
number-theoryarithmetic
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 additionally detects Nonstationarity:adf_pvalue.
What standard analysis sees
tail heaviness0.49
asymmetry0.05
occupancy0.55
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
Nonstationarity:adf_pvalue+3.0z
unit-root nonstationarity (ADF cannot reject random-walk null)
Atlas-extreme metrics the standard bank can’t predict for this source
H3 Icosahedral:temporal_coherence | +3.2z | bank-miss 3.1σ |
Möbius-S³:phi_return_cv | +2.9z | bank-miss 1.8σ |
H3 Icosahedral:nn_enrichment | +2.9z | bank-miss 2.9σ |
Chladni:nodal_clustering | +2.5z | bank-miss 1.1σ |
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 > free_plate_nodal
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 | |
|---|---|---|
| Hilbert Walk | 2.92 | cross-origin |
| Rudin-Shapiro | 3.14 | cross-origin |
| Brownian Walk | 3.39 | cross-origin |
Which Geometries Light Up
Sol (Thurston) › Sol (Thurston):integrated_walk_smoothness | rank 5/307 | 0.8879 |
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