The Divisor Function along a Deterministic Orbit and the Emergence of Ladders
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arXiv
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866914513833426944 |
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| author | Mantovanelli, Marco |
| author_facet | Mantovanelli, Marco |
| contents | We study the deterministic recursion $n_{j+1} = n_j - τ(n_j)$, where $τ(n)$ denotes the divisor function, and the associated orbit length $a(x)$. Heuristics based on the average order of $τ(n)$ suggest that $a(x) \asymp x / \log x$, but the strong dependence along the orbit places the problem outside the scope of existing methods for multiplicative functions. We develop a deterministic framework that reduces the analysis of the orbit to the distribution of $τ(n_j)$ on dyadic scales. This yields a structure-versus-randomness principle: either the orbit exhibits divisor mixing, or it develops strong additive structure. In the latter case, we show, under a phase-rigidity hypothesis, that the orbit contains long near-arithmetic progressions along which $τ(n)$ is essentially constant, which we call divisor ladders. Our main result reduces the asymptotic behavior of $a(x)$ to a single structural obstruction. Assuming an anti-concentration hypothesis that rules out energy-saturating divisor ladders, we obtain $a(x) \asymp x / \log x$. The paper also establishes several unconditional structural results, including that large values of $τ(n)$ are negligible on dyadic scales and that any potential obstruction must occur at a single divisor scale $τ(n) \asymp \log n$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_25446 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The Divisor Function along a Deterministic Orbit and the Emergence of Ladders Mantovanelli, Marco Number Theory Combinatorics Dynamical Systems Primary: 11N37 Secondary: 11A25, 11N60, 37C10 We study the deterministic recursion $n_{j+1} = n_j - τ(n_j)$, where $τ(n)$ denotes the divisor function, and the associated orbit length $a(x)$. Heuristics based on the average order of $τ(n)$ suggest that $a(x) \asymp x / \log x$, but the strong dependence along the orbit places the problem outside the scope of existing methods for multiplicative functions. We develop a deterministic framework that reduces the analysis of the orbit to the distribution of $τ(n_j)$ on dyadic scales. This yields a structure-versus-randomness principle: either the orbit exhibits divisor mixing, or it develops strong additive structure. In the latter case, we show, under a phase-rigidity hypothesis, that the orbit contains long near-arithmetic progressions along which $τ(n)$ is essentially constant, which we call divisor ladders. Our main result reduces the asymptotic behavior of $a(x)$ to a single structural obstruction. Assuming an anti-concentration hypothesis that rules out energy-saturating divisor ladders, we obtain $a(x) \asymp x / \log x$. The paper also establishes several unconditional structural results, including that large values of $τ(n)$ are negligible on dyadic scales and that any potential obstruction must occur at a single divisor scale $τ(n) \asymp \log n$. |
| title | The Divisor Function along a Deterministic Orbit and the Emergence of Ladders |
| topic | Number Theory Combinatorics Dynamical Systems Primary: 11N37 Secondary: 11A25, 11N60, 37C10 |
| url | https://arxiv.org/abs/2604.25446 |