von Mangoldt Function
number-theory · 37 views
number-theoryarithmetic
What It Is
von Mangoldt Lambda(n) = log(p) if n = p^k for some prime p and k >= 1, else 0. Prime-power-supported sparse signal whose cumulative sum is the Chebyshev psi function. By Riemann's explicit formula psi(x) = x - sum_rho x^rho/rho - log(2 pi) - (1/2) log(1 - x^-2), so Lambda encodes the nontrivial zeros of zeta directly via psi --- the closest one gets to the zeta zeros in pure number-theoretic form.
Interpretation
Standard analysis sees: heavy-tailed; right-skewed; few distinct values; low-complexity (predictable, not noise-like); homoskedastic; multifractal; low-dimensional. The atlas additionally detects discrete-map sensitive dependence.
What standard analysis sees
tail heaviness0.87
asymmetry0.95
occupancy0.01
short-range corr0.16
long-range memory0.18
spectral colour0.84
periodicity0.62
complexity0.12
time-irreversibility0.45
volatility clustering0.08
multifractality0.91
dimensionality0.01
nonstationarity0.41
What the atlas adds
discrete-map sensitive dependence+5.6z
deterministic chaos (positive λ_max, sensitive dependence)
names a discrete-map-scoped estimate, NOT chaos in general — continuous-flow chaos (Lorenz, Rössler) reads weak/neutral here despite being genuinely chaotic; spiky arithmetic sources can false-positive on the finite-time estimateAtlas-extreme metrics the standard bank can’t predict for this source
Ulam Spiral (Square):diagonal_alignment | +4.4z | bank-miss 1.6σ |
Higher-Order Statistics:skew_mean | +3.9z | bank-miss 1.1σ |
Zipf–Mandelbrot (8-bit):hapax_ratio | +3.7z | bank-miss 4.0σ |
Zipf–Mandelbrot (16-bit):zipf_alpha | +3.7z | bank-miss 1.4σ |
Nonstationarity:change_quantiles_high | +3.4z | bank-miss 2.7σ |
Hodge–Laplacian:source_fraction | -3.1z | bank-miss 1.8σ |
Zipf–Mandelbrot (8-bit):zipf_alpha | +2.7z | bank-miss 3.7σ |
Mostow Rigidity:spectral_rigidity | -2.5z | bank-miss 1.9σ |
Composition
dtypefloat64
range[0, 11.53]
unique values1439 / 16384
mean ± std1 ± 3.23
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/von_Mangoldt_Function.png)
Heisenberg (Nil) (centered) > signed_log_z
_(centered)/xy_path/von_Mangoldt_Function.png)
Heisenberg (Nil) (centered) > xy_path
Hodge–Laplacian > default
/barcode/von_Mangoldt_Function.png)
Inflation (Substitution) > barcode
/d_curve/von_Mangoldt_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/von_Mangoldt_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/von_Mangoldt_Function.png)
Ulam Spiral (Sacks) > default
/default/von_Mangoldt_Function.png)
Ulam Spiral (Square) > default
Visibility Graph > default
Wavelet Cascade > default
Atlas Position
| Nearest neighbor | Distance | |
|---|---|---|
| Prime Indicator | 3.13 | |
| Gap-Word β-Shift | 4.63 | cross-origin |
| Neural Net (Pruned 90%) | 4.68 | cross-origin |
Which Geometries Light Up
Attractor Reconstruction › Attractor Reconstruction:lyapunov_max | rank 3/307 | 0.5361 |
Higher-Order Statistics › Higher-Order Statistics:skew_mean | rank 5/307 | 2.5552 |
Hyperbolic (Poincaré) › Hyperbolic (Poincaré):curvature_structure | rank 4/307 | 575.5498 |
Level Statistics › Level Statistics:spacing_gue_distance | rank 5/307 | 0.9045 |
Level Statistics › Level Statistics:spacing_poisson_distance | rank 5/307 | 0.9045 |
Moiré › Moiré:moire_peak_alpha | rank 5/307 | 3.4000 |
Nonstationarity › Nonstationarity:dynamic_coupling | rank 4/307 | 9.2547 |
Zipf–Mandelbrot (16-bit) › Zipf–Mandelbrot (16-bit):zipf_alpha | rank 2/307 | 2.5923 |
Zipf–Mandelbrot (8-bit) › Zipf–Mandelbrot (8-bit):hapax_ratio | rank 2/307 | 0.3496 |
Zipf–Mandelbrot (8-bit) › Zipf–Mandelbrot (8-bit):zipf_alpha | rank 4/307 | 3.1801 |
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