Murmurations of Dirichlet characters

Stage M2 — Lee–Oliver–Pozdnyakov (arXiv:2307.00256), Theorem 1.1. The Gauss-sum-normalized average χ(p)/τ(χ) over primitive even/odd characters, ordered by prime conductor, converges to an exact density: sharp interval → cos(2πy) (even) / −i·sin(2πy) (odd). Zero external data — characters, Gauss sums, and primes are all computed from scratch. Renders emitted JSON only.

Sharp interval: P̃₊ (even, real) vs cos(2πy), and P̃₋ (odd, imaginary) vs −sin(2πy)

P̃₊ even (Re) cos(2πy) P̃₋ odd (Im) −sin(2πy)

Sharp interval: P̃₊ (even, real) vs cos(2πy), and P̃₋ (odd, imaginary) vs −sin(2πy)

Convergence to the density (observed empiric) & the parity / null structure