The Divisor Function along a Deterministic Orbit and the Emergence of Ladders

Abstract

We study the deterministic recursion $n_{j+1} = n_j - \tau(n_j)$, where $\tau(n)$ denotes the divisor function, and the associated orbit length $a(x)$. Heuristics based on the average order of $\tau(n)$ 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 $\tau(n_j)$ 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 $\tau(n)$ 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 $\tau(n)$ are negligible on dyadic scales and that any potential obstruction must occur at a single divisor scale $\tau(n) \asymp \log n$.