A family of analogues to the Robin criterion
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866917057852866560 |
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| author | Fan, Steve Kobayashi, Mits Molnar, Grant |
| author_facet | Fan, Steve Kobayashi, Mits Molnar, Grant |
| contents | The Robin criterion states that the Riemann hypothesis is equivalent to the inequality $σ(n) < e^γn \log \log n$ for all $n>5040$, where $σ(n)$ is the sum of divisors of $n$, and $γ$ is the Euler--Mascheroni constant. Define the family of functions \[ σ^{[k]} (n):=\sum_{[d_1,\dots,d_k]=n}d_1\dots d_k \] where $[d_1, \dots, d_k]$ is the least common multiple of $d_1, \dots, d_k$. These functions behave asymptotically like $σ(n)^k$ as $k\to\infty$. We prove the following analogue of the Robin criterion: for any $k \geq 2$, the Riemann hypothesis holds if and only if $σ^{[k]} (n) < \frac{(e^γn \log \log n)^k}{ζ(k)}$ for all $n > 2162160$, where $ζ$ is the Riemann zeta function. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_02106 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A family of analogues to the Robin criterion Fan, Steve Kobayashi, Mits Molnar, Grant Number Theory 11N37, 11A25, 11M26 The Robin criterion states that the Riemann hypothesis is equivalent to the inequality $σ(n) < e^γn \log \log n$ for all $n>5040$, where $σ(n)$ is the sum of divisors of $n$, and $γ$ is the Euler--Mascheroni constant. Define the family of functions \[ σ^{[k]} (n):=\sum_{[d_1,\dots,d_k]=n}d_1\dots d_k \] where $[d_1, \dots, d_k]$ is the least common multiple of $d_1, \dots, d_k$. These functions behave asymptotically like $σ(n)^k$ as $k\to\infty$. We prove the following analogue of the Robin criterion: for any $k \geq 2$, the Riemann hypothesis holds if and only if $σ^{[k]} (n) < \frac{(e^γn \log \log n)^k}{ζ(k)}$ for all $n > 2162160$, where $ζ$ is the Riemann zeta function. |
| title | A family of analogues to the Robin criterion |
| topic | Number Theory 11N37, 11A25, 11M26 |
| url | https://arxiv.org/abs/2511.02106 |