Liouville Function

What It Is

Liouville lambda(n) = (-1)^Omega(n) --- parity of prime-factor count with multiplicity. Pure +/-1 sequence with P(+1) -> 1/2 (Landau). Sum lambda(n) = O(n^(1/2+eps)) iff the Riemann Hypothesis; the Polya conjecture (false at n=906_150_257) probed the same balance.

Interpretation

Standard analysis sees: bounded / light-tailed; few distinct values; aperiodic / broadband; monofractal; stationary. The atlas finds no named structure, but the source is distinctively extreme on Möbius-S³:sector_transition_entropy (-2.6z) — beyond what the standard bank predicts for it. It sits beside Symbolic Henon in the atlas (standard-bank rank 20) — a neighbor conventional features miss.

Composition

Binary sequence — two distinct symbols.

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

Heisenberg (Nil) (centered) > signed_log_z

Heisenberg (Nil) (centered) > xy_path

Hodge–Laplacian > default

Inflation (Substitution) > barcode

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

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

Ulam Spiral (Sacks) > default

Ulam Spiral (Square) > default

Visibility Graph > default

Wavelet Cascade > default

Atlas Position

Open in Atlas →

Which Geometries Light Up