MathDB

Chowla conjecture

Let λ(n)=(−1)Ω(n)\lambda(n)=(-1)^{\Omega(n)} be the Liouville function. For every pair of distinct fixed nonnegative integers h1,h2h_1,h_2, one has lim⁡N→∞1N∑n≤Nλ(n+h1)λ(n+h2)=0\displaystyle \lim_{N\to\infty}\frac{1}{N}\sum_{n\le N}\lambda(n+h_1)\lambda(n+h_2)=0.

References

Primary source

arXiv

Additional references

analytic-number-theorymultiplicative-number-theory
Posted

Progress summary

Refreshed
Claimed progress

A new estimate strengthens evidence that prime-factor parity correlations cancel across growing shifts, but the original conjecture is not proved.

The Chowla conjecture predicts cancellation of correlations between Liouville values at distinct shifts, including the ordinary two-point Cesàro average. The central ordinary statement remains stronger than all logarithmically averaged results reported here.

Known results

  • Tao (2015): proved the logarithmically averaged two-point result, including correlations along distinct affine shifts.
  • Helfgott and Radziwiłł (2022): obtained a major quantitative improvement for the logarithmic formulation using expander graphs.
  • Pilatte (2023): improved logarithmic savings and bounds for unweighted correlations at almost all scales.
  • Matomäki, Radziwiłł, and Tao (2026): improved short-interval bounds for an averaged Chowla formulation.

August 2026 growing-shift estimate

On August 24, 2026, a report on Quantitative Logarithmic Chowla Correlations Uniformly over Growing Shifts described an explicit estimate for logarithmically weighted two-point Liouville correlations over a growing shift range. This is claimed quantitative progress, but the preprint explicitly does not prove the ordinary Cesàro two-point conjecture.

Current status (as of August 2026): the ordinary Cesàro two-point Chowla conjecture remains open, while logarithmically averaged and other averaged forms have substantial quantitative results.

Sources

Solutions 0

No solutions have been posted yet.