Problem 1144

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claims/: The 1 claim page of Problem 1144, one per claimant's result; the problem's standing derives from them.

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Statement. Let $f$ be a random completely multiplicative function, where for each prime $p$ we independently choose $f(p)\in \{-1,1\}$ uniformly at random. Is it true that

$$\limsup_{N\to \infty}\frac{\sum_{m\leq N}f(m)}{\sqrt{N}}=\infty$$

with probability $1$?

Status. Claimed: one pending full claim, not accepted. The site labels the problem OPEN (page last edited 2026-01-26); the claim, filed on its proof-claims tab, is recorded here and not adopted: Høystad 2026, registered on 2026-09-06 with a write-up and a Lean 4 repository, which asserts the answer yes and credits GPT 6 Astra, GPT 5.6 Sol Pro and Fable 5. The repository reports its own axiom audit; this corpus has not built or audited the development, and no review or acceptance of the claim is recorded.

Source. erdosproblems.com/1144, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #1144, https://www.erdosproblems.com/1144.

References.

• [At25] C. Atherfold, Almost sure bounds for weighted sums of Rademacher random multiplicative functions. arXiv:2501.11076 (2025).

Formalization. None recorded.

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Linked library material

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• atherfold_2025_almost_sure_bounds_weighted_sums_rademacher