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Zubrilina's Formula

The mathematical proof that murmurations are real

Mr(X, P) = cr · Ai(α(P - P₀)) + O(P-δ)

The murmuration average M for rank r curves asymptotically follows an Airy-type function, where cr depends on rank, and α, P₀ are explicit constants.

Theoretical Prediction
Observed Data
Theory vs Data Overlay
Zubrilina's Formula
Rank 0 Data
Rank 1 Data

The Proof

Nina Zubrilina provided the first rigorous mathematical proof explaining the murmuration phenomenon. Her formula gives an explicit asymptotic expansion.

"Using very sophisticated math, she has proven an exact formula." — Peter Sarnak

Airy Functions

The appearance of Airy functions connects number theory to physics—these same functions describe light diffraction and quantum tunneling. A profound bridge between disciplines.

Universal Behavior

Zubrilina's work shows murmurations aren't accidents but arise from deep properties of L-functions. The same patterns appear across elliptic curves, modular forms, and Dirichlet characters.