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Robin's Inequality at Colossally Abundant Numbers and Nicolas' Inequality at Their Largest Prime Factor

Primarily AI-generated textHuman understanding: all partsmath.NT — Number Theorymath.NA — Numerical Analysis

Contributed by Frank Vega

Version 2 / Nov 02, 2026 / CC BY 4.0

Abstract

Robin's criterion states that the Riemann Hypothesis holds if and only if σ(n)<eγ⋅n⋅log⁡log⁡n\sigma(n) < e^{\gamma} \cdot n \cdot \log\log n for every n>5040n > 5040, and Nicolas' criterion states that it holds if and only if c(q)=log⁡(q#/φ(q#))−γ−log⁡log⁡θ(q)>0c(q) = \log(q\#/\varphi(q\#)) - \gamma - \log\log\theta(q) > 0 for every prime qq, where q#q\# is the primorial of qq and θ\theta is the Chebyshev function. Let NN be a colossally abundant number, let ξ(N)\xi(N) be the real number associated with NN by Nicolas, and let qq be the largest prime not exceeding ξ(N)\xi(N). Writing Δ(N)=log⁡(σ(N)/N)−γ−log⁡log⁡log⁡N\Delta(N) = \log(\sigma(N)/N) - \gamma - \log\log\log N, we prove without any hypothesis that, whenever log⁡q≥31\log q \geq 31, 1.353/(qlog⁡2q)≤Δ(N)−c(q)+22/(qlog⁡q)≤3.970/(qlog⁡2q)1.353/(\sqrt{q} \log^{2} q) \leq \Delta(N) - c(q) + 2\sqrt{2}/(\sqrt{q} \log q) \leq 3.970/(\sqrt{q} \log^{2} q). The range log⁡q≥31\log q \geq 31 contains the prime qq of every colossally abundant counterexample, in accordance with the author's result that the largest prime factor of the least counterexample exceeds e31.018e^{31.018}. Thus the functions Δ(N)\Delta(N) and c(q)c(q) differ by 22/(qlog⁡q)2\sqrt{2}/(\sqrt{q} \log q) up to an explicit error of order 1/(qlog⁡2q)1/(\sqrt{q} \log^{2} q). This comparison is two-sided, but it is not an equivalence between Robin's inequality at NN and Nicolas' inequality at qq: it does not decide Robin's inequality at NN when c(q)qlog⁡qc(q) \sqrt{q} \log q lies between 22(1−1.4035/log⁡q)2\sqrt{2}(1 - 1.4035/\log q) and 22(1−0.4787/log⁡q)2\sqrt{2}(1 - 0.4787/\log q). As consequences, every colossally abundant counterexample to Robin's inequality yields a prime qq with c(q)>2.70/(qlog⁡q)c(q) > 2.70/(\sqrt{q} \log q); conversely, every prime qq with log⁡q≥31\log q \geq 31 and c(q)>22(1−0.4787/log⁡q)/(qlog⁡q)c(q) > 2\sqrt{2}(1 - 0.4787/\log q)/(\sqrt{q} \log q) yields a colossally abundant counterexample, which gives a derivation of Robin's known theorem on colossally abundant counterexamples from Nicolas' oscillation theorem; and under the Riemann Hypothesis every colossally abundant number with log⁡q≥31\log q \geq 31 satisfies Δ(N)<−0.654/(qlog⁡q)\Delta(N) < -0.654/(\sqrt{q} \log q), in accordance with the constant 2(2−1)2(\sqrt{2}-1) of Ramanujan's bound. For superabundant counterexamples, using the exponent bound of Nazardonyavi and Yakubovich, which gives log⁡n≥ψ(pk)\log n \geq \psi(p_{k}) for the largest prime factor pkp_{k}, we prove c(pk)>0.9991/(pklog⁡pk)c(p_{k}) > 0.9991/(\sqrt{p_{k}} \log p_{k}), a bound that applies to the least counterexample, which is superabundant, and we show that the least colossally abundant number not smaller than the least counterexample nn is itself a counterexample NN with log⁡N−log⁡ξ(N)≤log⁡n\log N - \log\xi(N) \leq \log n.

Author comments: In the revised version we emphasize that Theorem 1.1 is unconditional. Nicolas’ estimates in [20] are proved under RH and, under RH, give a narrower interval than ours (Remark 3.5). Since a counterexample to Robin’s inequality exists only if RH is false, an estimate that assumes RH cannot be applied to one, whereas Theorem 1.1 applies to every colossally abundant number with log q ≥ 31. This is what Corollaries 4.2–4.4 use.

Provenance statement

The author used Claude Opus 5.5 to write the verification script verify_robin_nicolas.py of Section 6 and to generate parts of the text of this manuscript. The experiments were run by the author. After using this tool, the author reviewed and edited the code and the content as needed and takes full responsibility for the content of the publication.
AI provenance filesverify_robin_nicolas.py · 9 KB

Tools used

Anthropic
Claude OpusVersion 5.5

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Version history

  1. v1 · 14 pagesSubmitted by Frank VegaInitial depositSupersededOct 31, 2026
  2. v2 · 17 pagesSubmitted by Frank VegaIn the revised version we emphasize that Theorem 1.1 is unconditional. Nicolas’ estimates in [20] are proved under RH and, under RH, give a narrower interval than ours (Remark 3.5). Since a counterexample to Robin’s inequality exists only if RH is false, an estimate that assumes RH cannot be applied to one, whereas Theorem 1.1 applies to every colossally abundant number with log q ≥ 31. This is what Corollaries 4.2–4.4 use.CurrentNov 10, 2026