On the Lagarias Inequality and Superabundant Numbers
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arXiv
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866911456498286592 |
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| author | MacArevey, Andrew |
| author_facet | MacArevey, Andrew |
| contents | We study the Lagarias inequality, an elementary criterion equivalent to the Riemann Hypothesis. Using a continuous extension of the harmonic numbers, we show that the sequence $B_n=\frac{H_n+e^{H_n}\log(H_n)}{n}$ is strictly increasing for $n\ge 1$. As a consequence, if the Lagarias inequality has counterexamples, then the least counterexample must be a superabundant number; equivalently, it suffices to verify the inequality on the superabundant numbers. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_15905 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the Lagarias Inequality and Superabundant Numbers MacArevey, Andrew Number Theory We study the Lagarias inequality, an elementary criterion equivalent to the Riemann Hypothesis. Using a continuous extension of the harmonic numbers, we show that the sequence $B_n=\frac{H_n+e^{H_n}\log(H_n)}{n}$ is strictly increasing for $n\ge 1$. As a consequence, if the Lagarias inequality has counterexamples, then the least counterexample must be a superabundant number; equivalently, it suffices to verify the inequality on the superabundant numbers. |
| title | On the Lagarias Inequality and Superabundant Numbers |
| topic | Number Theory |
| url | https://arxiv.org/abs/2602.15905 |