Logarithmically averaged Chowla conjecture for even k ≥ 4: Σ λ(n)λ(n+1)λ(n+2)λ(n+3)/n = o(log x)
The odd cases fell in 2017 and uniformity is known for H ≥ X^θ, so the gap is the slowly growing range of H; that makes it easier than breaking parity for prime gaps (80) and comparable to sunflowers (71). Application value 3 is minimal. Leverage 42: equivalent to logarithmic Sarnak and to local Gowers uniformity of λ, which many dynamical and sign-pattern results would use.
The problem
Show that the signs of whole numbers' prime-factor counts at four nearby positions look random together.
- 1Sign each number by prime-factor parity
- 2Signs behave like fair coin flips
- 3Chowla: products at nearby spots average zero
- 4Two and odd counts proved, four unproved
The Liouville function , where is the number of prime factors of counted with multiplicity, takes values . It is closely tied to the Möbius function and to primes: is equivalent to the prime number theorem. The says that for distinct shifts , the correlation . The version replaces the plain average with weights :
Path to understanding
foundations → problemThe research entries this problem rests on, in the order to read them. Each links to its explainer.
- Start with
- Harmonic analysisTheory
- Step 2
- Step 3
- The Riemann zeta function and prime distributionTheory · host
- Then
- Logarithmically averaged Chowla conjecture for even k ≥ 4: Σ λ(n)λ(n+1)λ(n+2)λ(n+3)/n = o(log x)The problem
State of the art
Open for even k ≥ 4. Tao (arXiv:1509.05422, 2016) proved the logarithmically averaged two-point case with the entropy decrement argument; Helfgott–Radziwiłł (2021) and Pilatte (arXiv:2310.19357) made it quantitative, Pilatte with a saving (log x)^(−c). Tao and Teräväinen (arXiv:1708.02610, Duke 2019; shorter proof arXiv:1710.02112) proved every odd order, where a parity symmetry kills the structured part, and deduced the predicted logarithmic density of all length-3 sign patterns of λ and length-4 patterns of μ. Matomäki–Radziwiłł–Tao (arXiv:1509.01545) showed earlier that all eight length-3 sign patterns of λ occur with positive lower natural density. For even k the obstruction is a possible correlation of λ with nilsequences on short intervals. Matomäki–Radziwiłł–Tao (arXiv:1812.01224) proved local Fourier uniformity on average for H ≥ X^θ, and Matomäki–Radziwiłł–Tao–Teräväinen–Ziegler (arXiv:2007.15644, Annals 2023) higher-order uniformity in the same range; the conjecture needs H growing arbitrarily slowly. Tao (arXiv:1605.04628) proved the equivalence of the logarithmic Chowla and Sarnak conjectures. OpenAI's 6 October 2026 release (github.com/openai/math, family 007; unrefereed, with a Lean formalization listed) claims the unweighted two-point case with a (log x)^(−c) saving; it does not address k ≥ 3 without weights or even k ≥ 4 with weights.
Realistic applications
None practical. The k-point Chowla conjecture underlies sign-pattern and prime-tuple heuristics and, through Sarnak’s conjecture, Möbius disjointness for zero-entropy dynamical systems. Methods (entropy decrement, short-interval uniformity of multiplicative functions) are reused in ergodic theory and additive combinatorics.
What counts as solved
A refereed proof of the logarithmically averaged Chowla conjecture for k = 4 (all distinct shifts), or of local Fourier uniformity of λ for H → ∞ arbitrarily slowly. Results needing H ≥ X^θ, or odd k, do not count.
How it gets verified
Expert reviewOnly expert review of the argument is possible today.
The target is the logarithmically averaged Chowla conjecture for k = 4, or local Fourier uniformity of the Liouville function. Mathlib has no Liouville-function correlation theory, and the proofs rest on entropy decrement and ergodic methods that are far from formalised. Referees check it; results needing H >= X^theta or odd k do not count.
Claims under verification
+ add a claimNo claimed results on the Verification board for this problem yet.
Where the work is
Purely analytic: the missing input is a short-interval inverse theorem for λ; computation only explores numerics of correlations.
Bottlenecks
ranked by votes- #1TheoryLocal uniformity at slowly growing scales
Known short-interval results for λ against polynomial phases and nilsequences need H ≥ X^θ; even-order Chowla needs H → ∞ arbitrarily slowly, where the Matomäki–Radziwiłł method gives no control.
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Connected problems
- This problem is related to Prime gaps beyond the sieve parity limit: H₁ < 6 under generalized Elliott–Halberstam
Progress
- Jul 30, 2020Higher-order uniformity for H ≥ X^θsource
Matomäki–Radziwiłł–Tao–Teräväinen–Ziegler (Annals 2023); the slowly growing range needed for even k remains open.
- Aug 8, 2017All odd orderssource
Tao–Teräväinen prove the logarithmic conjecture for every odd k (arXiv August 2017).
- Sep 17, 2015Log-averaged two-point Chowlasource
Tao’s entropy decrement argument.