Math Conjectures

Chowla’s Correlation Conjecture

OPENLandmarkConjectureProposed 1965 · Full conjecture

Canonical statement

Let λ(n)=(−1)Ω(n)\lambda(n)=(-1)^{\Omega(n)} be the Liouville function, where Ω(n)\Omega(n) is the number of prime factors of nn, counted with multiplicity. For every integer k≥2k\ge2 and every choice of distinct nonnegative integers h1,…,hkh_1,\ldots,h_k,
∑n≤x∏j=1kλ(n+hj)=o(x)(x→∞). \sum_{n\le x}\prod_{j=1}^k\lambda(n+h_j)=o(x) \qquad(x\to\infty).
View source LaTeX
Let \(\lambda(n)=(-1)^{\Omega(n)}\) be the Liouville function, where \(\Omega(n)\) is the number of prime factors of \(n\), counted with multiplicity. For every integer \(k\ge2\) and every choice of distinct nonnegative integers \(h_1,\ldots,h_k\),
\[
  \sum_{n\le x}\prod_{j=1}^k\lambda(n+h_j)=o(x)
  \qquad(x\to\infty).
\]
NT-CORR-035:[Chowla1965Riemann][Tao2016LogChowla][MatomakiRadziwillTao2023HigherUniformity]
Number theoryAnalysis and functional analysis

Notes

Logarithmically averaged two-point and higher-order variants, almost-all-shift results, and several conditional cases are known. Removing the logarithmic weight and proving the stated cancellation for every fixed tuple of distinct shifts remains open already for two shifts.

References (3)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.