Research notes on every problem, and a library of the papers behind them. Built from the open erdos repository.

Updated Oct 8

Problem 964

claims/: The 1 claim page of Problem 964, one per claimant's result; the problem's standing derives from them.

Statement. Let $\tau(n)$ count the number of divisors of $n$ . Is the sequence

\frac{\tau(n+1)}{\tau(n)}

everywhere dense in $(0,\infty)$ ?

Status. Proved. The site labels the problem PROVED (LEAN) and credits Eberhard's proof, which answers the question affirmatively; his stronger theorem says every positive rational occurs infinitely often. The community Lean file posted in the forum thread contains a complete formal proof conditional on a formalized GGPY proposition, which that file does not prove. The accepted claim is recorded on Eberhard's claim page.

Source. erdosproblems.com/964, accessed 2026-09-05. Cite as: T. F. Bloom, Erdős Problem #964, https://www.erdosproblems.com/964, accessed 2026-09-05.

References.

[Eb25] S. Eberhard, Ratios of consecutive values of the divisor function. Journal of Number Theory 281 (2026), 426--428. DOI: https://doi.org/10.1016/j.jnt.2025.10.002. Preprint: https://arxiv.org/abs/2505.00727.
[Er86b] P. Erdős, Some problems on number theory, Proceedings of the Seventeenth Southeastern International Conference on Combinatorics, Graph Theory, and Computing (1986), 225--244.

Formalization. No Formal Conjectures statement is recorded by the site. Daniel Chin posted the community Lean file in #post-4280: https://github.com/danielchin/proofs/blob/e33cb5565aef08a2ebb430beba6c84644fe7fca4/Proofs/ErdosProblems/Erdos964.lean (pinned to the commit of 14 February 2026 that last changed the file). A post of the same day, #post-4291, quotes Terry Tao's assessment that the proof is conditional on GGPY and appears correctly formalized. Its final theorem is parameterized by GoldstonGrahamPintzYildirimStatement, so this is a conditional formal proof of the downstream argument. The file declares no axiom and uses no sorry or admit; this corpus has not built it.

Current assessment

The affirmative status is supported here by Eberhard's published stronger theorem that every positive rational occurs infinitely often. The linked account transcribes that theorem and identifies the GGPY sieve input; its complete rewrite of Eberhard's published proof was checked against the paper by this corpus's own review, which is not acceptance evidence, with the GGPY sieve statement assumed at its recorded standing and not certified. No dated broader status search beyond erdosproblems.com is recorded. The community Lean file proves the downstream argument conditionally on GGPY, whose proposition is not proved in that file; this corpus has not built it. The problem's standing derives from Eberhard's claim page, accepted on the curator's credit and the journal publication; the community Lean file is recorded there as a formalization link, not as evidence.

The divisor-ratio review, grade and source reading filed with the Eberhard source read this page as it stood. On 2026-09-18T02:50:47Z the frontmatter status was changed from solved to proved and the Status field and one Known Results line were reworded to match, with the affirmative answer and Eberhard's theorem as its source unchanged; the Statement and this assessment were not touched by that edit.

Progress

The unconditional result is transcribed in [[../library/divisors/eberhard_2025_ratios_consecutive_values_divisor_function/main_theorem| Eberhard's main theorem]]. Its only external input is the GGPY two-of-three sieve result, stated with all hypotheses in [[../library/divisors/eberhard_2025_ratios_consecutive_values_divisor_function/theorem_1| Theorem 1]].

Known Results

Eberhard proves the stronger assertion that every $q\in\mathbb Q_{>0}$ is attained infinitely often by $\tau(n+1)/\tau(n)$ .
Schlage--Puchta, arXiv:2504.11463 (2025), gives quantitative bounds for related logarithmic ratio sets; it is background rather than an alternative proof of this problem.
Tao--Teräväinen, arXiv:2512.01739v2 (2026), §4.5, Remark 4.2, states a fixed-ratio local-limit asymptotic for $\tau(n+1)/\tau(n)=2^m a/b$ outside the exceptional set inherited from their Theorem 1.7, and says it can recover Eberhard's density result. The source leaves that fixed-ratio generalization to the reader; see [[../library/arithmetic_functions/tao_2025_quantitative_correlations_problems_prime_factors_consecutive/remark_4_2_divisor_ratio| the precise source record]].

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

Linked from (13)

Eberhard's every positive rational as a consecutive divisor-count ratio problems/divisors/E0964/claims Divisors and Multiples arithmetic_functions/tao_2025_quantitative_correlations_problems_prime_factors_consecutive Remark 4.2: fixed-ratio divisor local limit formula Divisors and Multiples divisors/eberhard_2025_ratios_consecutive_values_divisor_function Source readings and corrections for the divisor-ratio review Every positive rational ratio occurs infinitely often Theorem 1: two of three linear forms are E2-values The divisor function at consecutive integers Theorem 2 (p. 308): for positive integers d_1, ..., d_7, the counts of n <= x with d(n+1)/d(n) = d_j/d_i, summed over i < j, are at least c x (log log x)^{-3}Theorem 3 (p. 308): the limit points of log(d(n+1)/d(n)) have positive lower density on each half-line and fill an interval [-δ, δ]

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