Short··0:21·Number theory
Chowla Conjecture, Two-Point Case: OpenAI Claims a Proof
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About this video
Liouville's function gives each whole number +1 if it has an even number of prime factors and −1 if odd. Chowla's conjecture says neighbouring values are uncorrelated: the average of λ(n)λ(n+1) up to X tends to zero. OpenAI claims a proof of the ordinary two-point Chowla conjecture, with a power-of-logarithm saving for any two fixed nonproportional affine forms, and of the binary corrected Elliott conjecture for bounded multiplicative functions; this is OpenAI's preprint, not yet peer reviewed. Tao proved the logarithmically averaged version in 2015.
Paper (OpenAI, catalogue result 007):
- Ordinary two-point correlations of multiplicative functions: github.com/openai/math/blob/main/preprints/Ordinary-two-point…
The animation illustrates the result; it is not a proof.
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About the problem
Number theory·OpenAI Math·Result 007
Ordinary two-point correlations and the corrected Elliott conjecture
Proves the ordinary two-point Chowla conjecture, with a bound O(X/(log X)c) for Liouville correlation sums along fixed nonproportional affine forms, where c > 0 is absolute. More generally, proves the binary corrected Elliott conjecture for complex multiplicative functions bounded by one when one factor is uniformly nonpretentious against each fixed Dirichlet character times nit for |t| ≤ X. (Lean)
Topics
Chowla conjecturetwo-point Chowla conjectureElliott conjectureLiouville functionmultiplicative functionscorrelations of multiplicative functionsprime factorsanalytic number theoryTerence TaoSarnak conjecturenumber theoryprimesOpenAIOpenAI mathAI mathematicsAI proofmathsmathematicsmath shorts3D animation
