Ramanujan Tau
What It Is
Normalized Ramanujan tau τ̃(p) = τ(p)·p^(-11/2) for consecutive primes, where τ(n) is the q-coefficient of Δ(q) = η(q)^24 — the unique weight-12 cusp form on SL(2,ℤ). By Deligne (1974, formerly Ramanujan-Petersson), |τ̃(p)| ≤ 2 for all p; by Barnet-Lamb–Geraghty–Harris–Taylor (2011), {τ̃(p)} equidistributes to the Sato-Tate semicircle (2/π)√(1-x²/4) on [-2,2] — a non-Gaussian, non-uniform universal distribution distinct from GUE/Wigner spacing. Calibration anchor for metrics responding to semicircle-distributed sequences with deep automorphic structure.
Interpretation
Standard analysis sees: rich, high-entropy values; anti-persistent; aperiodic / broadband; high-complexity (noise-like); high-dimensional / space-filling. The atlas detects no named structure beyond this.
Nothing beyond the standard reading — this source’s structure is already captured by standard features; the atlas adds no named residual.
Composition
Render Gallery
_(centered)/signed_log_z/Ramanujan_Tau.png)
_(centered)/xy_path/Ramanujan_Tau.png)
/barcode/Ramanujan_Tau.png)
/d_curve/Ramanujan_Tau.png)
/phi_spectrum/Ramanujan_Tau.png)
/default/Ramanujan_Tau.png)
/default/Ramanujan_Tau.png)
Atlas Position
| Nearest neighbor | Distance | |
|---|---|---|
| ChaCha20 | 2.22 | cross-origin |
| glibc LCG | 2.28 | cross-origin |
| Wichmann-Hill | 2.39 | cross-origin |
Which Geometries Light Up
Cantor Set › Cantor Set:bit_plane_autocorrelation | rank 304/307 | 0.0024 |
Catch24 › Catch24:SB_MotifThree_quantile_hh | rank 1/307 | 2.1972 |
Catch24 › Catch24:SB_TransitionMatrix_3ac_sumdiagcov | rank 307/307 | 0.0000 |
D4 Triality › D4 Triality:gram_consistency | rank 1/307 | 2.5203 |
Heisenberg (Nil) (centered) › Heisenberg (Nil) (centered):area_length_ratio | rank 304/307 | 0.0000 |
Möbius-S³ › Möbius-S³:phase_profile_deviation | rank 305/307 | 0.1102 |
Predictability › Predictability:sample_entropy | rank 2/307 | 2.2412 |
S² × ℝ (Thurston) › S² × ℝ (Thurston):bingham_concentration | rank 306/307 | 0.4102 |