Ordinary two-point correlations of multiplicative functions
We prove the ordinary two-point Chowla conjecture. For every fixed pair of nonproportional affine forms, the Liouville correlation has a power-of-logarithm saving at every cutoff, with an absolute exponent. We also prove the binary corrected Elliott conjecture for ordinary averages of complex multiplicative functions of modulus at most one, under uniform nonpretentiousness of at least one original factor. This qualitative conclusion holds in fixed residue classes and for fixed nonproportional affine forms.
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In 1965, Chowla conjectured that the values of the Liouville function become asymptotically uncorrelated at consecutive natural numbers. Recently, the two-point case is resolved by OpenAI. In 2026, Charamaras and Richter conjectured a generalization of the Chowla conjecture to the asymptotic…
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Updates: Sharing AI progress on mathematics (amazing!); and my lecture plansCombinatorics and moreOct 7, 2026
announces the ordinary two-point Chowla conjecture, giving cancellation in the Liouville correlations (\sum_{n\le X}\lambda(n)\lambda(n+h)=o(X)) for every fixed (h\ge1). High-dimensional expanders. In High Dimensional Expanders: Introduction I (2011), I discussed several notions of expansion for…