The problem
For fixed distinct integers \(h_1, \dots, h_k\), prove that the average of \(\lambda(n+h_1)\cdots\lambda(n+h_k)\) over \(n \leq x\) tends to \(0\) as \(x \to \infty\), where \(\lambda\) is the Liouville function.
Chowla's Conjecture on Liouville Correlations
openHistory & significance
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.
Connected problems
- related The Lindelöf Hypothesis
- related Sarnak's Möbius Disjointness Conjecture
More in analytic number theory
- same area Oppermann's Conjecture
- same area Andrica's Conjecture
- same area Legendre's Conjecture
References
- sourceWikipedia
- sourceWikipedia — Liouville function
Still open.
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