OpenAI research collection · 1 manuscript · Source checked 2026-10-19
The idea in plain language
Count whether each integer has an even or odd number of prime factors, including repeats. This paper claims a precise cancellation result for paired patterns in that sequence, together with a wider result for related arithmetic functions.
Why it matters
It concerns rigorous forms of randomness within completely deterministic arithmetic, with detailed conditions in the manuscript.
Read the technical source claim
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 n^{it} for |t|≤ X.
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Short excerpts below come from OpenAI's abstracts. The full abstract and original PDF preserve the paper's precise scope.
MANUSCRIPT 01
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…
Full source abstract
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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New mathematical reasoning could help AI tackle harder scientific and computational problems. How reliably these skills transfer beyond mathematics will determine how much they contribute to general intelligence.