Riemann-Hardy-Littlewood

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: bounded / light-tailed; smooth / autocorrelated; long-range memory (persistent); strongly periodic; time-irreversible (sharp rises, slow decay); volatility-clustering (bursty); low-dimensional. The atlas additionally detects Nonstationarity:adf_pvalue.

Composition

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 →

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