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Robin's Inequality and Superabundant Counterexamples

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

Contributed by Frank Vega ↗

Version 1 / Oct 26, 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. Let n′n' be the squarefree kernel of nn, pkp_{k} its largest prime factor, and d=eγ/ζ(2)≈1.08276d = e^{\gamma}/\zeta(2) \approx 1.08276. We prove that Robin's inequality holds for every n>5040n > 5040 with d⋅log⁡log⁡n′≤log⁡log⁡nd \cdot \log\log n' \leq \log\log n, and for every n>5040n > 5040 whose prime factors are all below (log⁡n)1/d(\log n)^{1/d}, and that both results remain valid with dd replaced by 1.0161.016. Our main results concern superabundant numbers, among which there are infinitely many counterexamples to Robin's inequality if the Riemann Hypothesis is false. Using the classical error term in the prime number theorem, we show that every superabundant counterexample satisfies log⁡log⁡n−log⁡pk<C⋅exp⁡(−clog⁡pk)\log\log n - \log p_{k} < C \cdot \exp(-c \sqrt{\log p_{k}}) for absolute constants C,c>0C, c > 0. Since the least counterexample satisfies log⁡n>pk\log n > p_{k}, we deduce that the Riemann Hypothesis is true if there are constants c>0c > 0 and 0<α<20 < \alpha < 2 with c≥0.0562/31.032−αc \geq 0.0562/31.03^{2-\alpha}, and a threshold N≤101013.11485N \leq 10^{10^{13.11485}}, such that log⁡log⁡n−log⁡pk≥c/log⁡α(pk)\log\log n - \log p_{k} \geq c/\log^{\alpha}(p_{k}) for every superabundant number n>Nn > N with log⁡n>pk\log n > p_{k}. We also give explicit versions of these bounds: every counterexample satisfies pk1.000002>log⁡np_{k}^{1.000002} > \log n, and every superabundant counterexample satisfies −0.010101/(log⁡log⁡n)3<log⁡log⁡n−log⁡pk<0.056201/(log⁡log⁡n)2-0.010101/(\log\log n)^{3} < \log\log n - \log p_{k} < 0.056201/(\log\log n)^{2}. An enumeration of the superabundant numbers up to 104010^{40} shows that only 33 of the 244244 exceeding 50405040 satisfy e0.0562>1+(log⁡log⁡n−log⁡pk)⋅log⁡2(pk)e^{0.0562} > 1 + (\log\log n - \log p_{k}) \cdot \log^{2}(p_{k}), namely those with pk>log⁡np_{k} > \log n, the largest of which is 1068586291412640010685862914126400. Testing the hypothesis of the sufficient condition on the same range, we find that all 168168 superabundant numbers between 1068586291412640010685862914126400 and 104010^{40} satisfy log⁡log⁡n−log⁡pk>0\log\log n - \log p_{k} > 0, and that for each α∈{1,2,3,4,5}\alpha \in \{1, 2, 3, 4, 5\} the minimum of (log⁡log⁡n−log⁡pk)⋅log⁡α(pk)(\log\log n - \log p_{k}) \cdot \log^{\alpha}(p_{k}) over them is positive, ranging from 0.02230.0223 for α=1\alpha = 1 to 6.196.19 for α=5\alpha = 5, and is attained at the same superabundant number, whose largest prime factor is 5959.

Provenance statement

The author used Claude Opus 5.5 to write the verification script verify_robin_d.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_d.py · 12 KB

Tools used

Anthropic
Claude OpusVersion 5.5

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Version history

  1. v1Submitted by Frank VegaInitial depositCurrentOct 23, 2026