Chowla's Conjecture on Liouville Correlations

For fixed distinct integers \(h_{1}\),...,\(h_{k}\), prove that the average of λ(n+\(h_{1}\))⋯λ(n+\(h_{k}\)) over n ≤ x tends to 0 as x → ∞, where λ is the Liouville function.

Rooted in Pólya's 1919 urn model for prime races. Matomäki and Radziwiłł (2016) proved averages of λ over almost all short intervals are tiny — a revolution — and with Tao pushed the two-point correlation to o(x) for almost all shifts h via entropy decrement. The full k-point statement (even honest three-point) remains open; it sits just below parity-barrier technology.