MathDB

Chowla’s conjecture

For every integer k≥1k\ge 1 and every collection of distinct natural numbers h1,…,hkh_1,\ldots,h_k, where λ(n)=(−1)Ω(n)\lambda(n)=(-1)^{\Omega(n)} is the Liouville function, one has lim⁡N→∞1N∑n=1N∏j=1kλ(n+hj)=0\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}\prod_{j=1}^{k}\lambda(n+h_j)=0.

Sources & referencesView supporting material

Primary source

arXiv

Additional references

analytic-number-theorydynamical-systems

Progress summary

Refreshed
Partially solved

A new averaged dynamical theorem adds evidence, but the original fixed-shift conjecture remains open.

Chowla’s conjecture predicts that products of Liouville-function values at any fixed collection of distinct shifts average to zero. Sarvadaman Chowla formulated it in 1965.

Known results

  • Matomäki–Radziwiłł (2015; published 2016) showed that long clusters of equal Liouville values are rare.
  • Matomäki–Radziłł–Tao (2015) proved an averaged form in which the shifts are averaged, not fixed.
  • Tao proved logarithmically averaged Chowla correlations; Helfgott–Radziwiłł later strengthened the quantitative range (2022).
  • A conditional 2024 result gives nontrivial fixed-shift bounds assuming a Landau–Siegel zero; no unconditional proof is known.

August 2026 dynamical generalization

Biao Wang proved an averaged dynamical analogue of Chowla-type sign independence, including an analogous result along the primes. The paper explicitly says that the original pointwise conjecture remains open; its corresponding non-averaged dynamical statement would imply Chowla’s conjecture.

Current status (as of August 2026): Averaged and logarithmically averaged forms, plus conditional and dynamical analogues, are established, but Chowla’s original fixed-shift pointwise conjecture remains open.

Sources

Solutions 0

No solutions have been posted yet.