Eberhard 2025 ratios consecutive values divisor function

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main_theorem: Eberhard proves that every positive rational occurs infinitely often as a ratio of consecutive values of the divisor function.

theorem_1: A special case of the GGPY sieve gives infinitely many simultaneous two-almost-prime values among three suitably coprime linear forms.

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Sean 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.

The paper proves unconditionally that every positive rational equals $d(n+1)/d(n)$ for infinitely many $n$. This is stronger than the original density question. Its proof takes the set $R$ of values attained infinitely often and uses the special case $b_1=b_2=b_3=1$ of the Goldston--Graham--Pintz--Yıldırım sieve (their Corollary 2.1) to put one of three explicit divisor ratios in $R$. Multiplying the auxiliary $r_i$ by carefully chosen prime powers equalizes the three candidates. A parametrization by distinct prime blocks then puts in $R$ every finite product of the values

$$f(x,y)=\frac{(x+1)(y+1)}{x+y+1}\qquad(x,y\geq1)$$

and their inverses, so $R$ contains the subgroup of $\mathbb Q_{>0}$ that these values generate. That subgroup is all of $\mathbb Q_{>0}$: $2=f(2,3)$, and for $p=2x+1>2$ prime, $p=(x+1)^2/f(x,x)$. The full rewritten proof is in Every positive rational ratio occurs infinitely often. The external sieve input is stated with its hypotheses in Theorem 1.

The canonical file in this folder is the published three-page PDF, retrieved from https://wrap.warwick.ac.uk/id/eprint/194323/7/1-s2.0-S0022314X25002926-main.pdf. The accepted arXiv v2 (30 October 2025) is retained as eberhard_2025_ratios_consecutive_values_divisor_function_arxiv_v2.pdf, and arXiv v1 (27 April 2025) is retained as eberhard_2025_ratios_consecutive_values_divisor_function_arxiv_v1.pdf. The arXiv record and both retained versions were accessed at https://arxiv.org/abs/2505.00727. All three PDFs render cleanly; the published version and arXiv v2 have three pages, while v1 has two pages and omits the explicit equalizing exponents. The published file, eberhard_2025_ratios_consecutive_values_divisor_function.pdf, prints "0022-314X/© 2025 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/)." on its first page, the Creative Commons Attribution 4.0 license. For eberhard_2025_ratios_consecutive_values_divisor_function_arxiv_v1.pdf the arXiv record names the Creative Commons Attribution 4.0 license (arXiv:2505.00727). For eberhard_2025_ratios_consecutive_values_divisor_function_arxiv_v2.pdf the same arXiv record names the Creative Commons Attribution 4.0 license (arXiv:2505.00727).

The original arXiv record is https://arxiv.org/abs/2505.00727. The journal article identifies the prediction as Erdős [Erd86] and the method as building on Hasanalizade [Has21] (also discussed by Schlage-Puchta [Sch25]).

Bears on. #964

Results to transcribe.

• Main theorem: Every positive rational is attained infinitely many times by the sequence of ratios $d(n+1)/d(n)$.
• Theorem 1 (quoted GGPY input): Under the stated coprimality conditions, two of three linear forms divided by $r_i$ are products of two distinct primes above any prescribed bound $C$ for infinitely many inputs.

Related work

• J.-C. Schlage-Puchta, On a problem by Erdős and Mirsky on the ratio of the number of divisors of consecutive integers, arXiv:2504.11463 (2025), https://arxiv.org/abs/2504.11463. This gives quantitative bounds for the logarithmic ratio closure functions defined there, including $L_+(x),L_-(x)\geq x/2$ and $\limsup_{x\to\infty}\min(L_+(x),L_-(x))/x\geq2/3$. It is a March 2025 historical predecessor; Eberhard's theorem supersedes it qualitatively by giving every positive rational ratio infinitely often.
• T. Tao and J. Teräväinen, Quantitative correlations and some problems on prime factors of consecutive integers, arXiv:2512.01739v2 (25 April 2026), [[arithmetic_functions/tao_2025_quantitative_correlations_problems_prime_factors_consecutive/remark_4_2_divisor_ratio| §4.5, Remark 4.2]], https://arxiv.org/html/2512.01739v2. For a fixed rational $a/b>0$ with odd numerator and denominator, the first bullet of that remark states, uniformly for integers $m$ and outside the exceptional set of $x$ inherited from their Theorem 1.7, where $\log_2 x=\log\log x$,

$$\frac{1}{x}\left|\left\{n\leq x:\frac{\tau(n+1)}{\tau(n)} =2^m\frac{a}{b}\right\}\right| =\frac{c_{\tau,a/b}e^{-m^2/(4\log_2 x)} +O(\log_2^{-c}x)}{2\sqrt{\pi\log_2 x}}.$$

They say this can recover Eberhard's density result, while leaving these fixed-ratio generalizations to the reader. The exact statement is recorded in [[arithmetic_functions/tao_2025_quantitative_correlations_problems_prime_factors_consecutive/remark_4_2_divisor_ratio| Remark 4.2]]. It is a materially distinct quantitative local-limit approach; its fixed-ratio derivation is not supplied by the source and is not compiled here, so proof coverage remains incomplete for that result.