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Two-point Chowla conjecture for the truncated Möbius function over \mathbb F_q[t]

Let qq be fixed, let a∈Fq[t]∖{0}a\in\mathbb{F}_q[t]\setminus\{0\}, and let M=M(n)M=M(n) be in the truncation range specified for the truncated Möbius function. For a monic polynomial ff, write ωM(f)\omega_M(f) for the number of distinct irreducible divisors of ff retained by the truncation, and define μM(f)=(−1)ωM(f)\mu_M(f)=(-1)^{\omega_M(f)}. The two-point Chowla conjecture asserts that

lim⁡n→∞1qn∑f∈Fq[t] monicdeg⁡f=nμM(f) μM(f+a)=0.\lim_{n\to\infty}\frac{1}{q^n}\sum_{\substack{f\in\mathbb{F}_q[t]\text{ monic}\\ \deg f=n}}\mu_M(f)\,\mu_M(f+a)=0.
Equivalent formulations 1Other wordings

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

  1. Bivariate Erdős–Kac formulation

    The correlation problem is reformulated as the study of the joint distribution of (ωM(f),ωM(f+a))\bigl(\omega_M(f),\omega_M(f+a)\bigr), equivalently (ωM,ωM∘Ta)\bigl(\omega_M,\omega_M\circ T_a\bigr) for the translation Ta(f)=f+aT_a(f)=f+a.

    source: Chowla Conjecture for the truncated Möbius function over $\mathbb{F}_q[t]$

References

Primary source

arXiv

Additional references

analytic-number-theoryfunction-fieldssieve-theory
Posted · edited

Progress summary

Refreshed
Claimed solved

An unrefereed preprint claims to prove the two-point cancellation statement for a truncated Möbius function over a function field, but no independent verification was found.

The problem asks for Chowla-type cancellation for the truncated Möbius function over Fq[t]\mathbb{F}_q[t]. It is a function-field result and does not imply the classical integer Chowla conjecture.

October 2026 claimed proof

Abhirup Chatterjee's unrefereed preprint claims the required correlation estimate, using a bivariate Erdős–Kac theorem together with a pure Brun sieve. This would establish the stated two-point result in the fixed-characteristic function-field setting, but the claim is unverified.

Current status (as of October 2026): The function-field statement is claimed proved by an unrefereed preprint, but its correctness has not been independently verified; the classical integer conjecture remains open.

Sources

Solutions 0

No solutions have been posted yet.