The Subtractive Divisor Orbit: Unconditional Bounds, Parity Constraints, and a Conditional Framework

2026/04/28 by Marco Mantovanelli
#math.NT #math.CO #math.DS

paper · pdf

Abstract

Let τ(n) denote the number of positive divisors of n. Starting from n₀=x, consider the orbit n_j+1=n_j-τ(n_j), and let a(x) be its hitting time of zero. Although the average order of τ suggests a(x)\asymp x/log x, the orbit samples the divisor function endogenously, and no unconditional estimate of this order is known to us. We prove the exact identity ∑_j<a(x)τ(n_j)=x and the unconditional bounds (x)/((log(2x))³)≪ a(x)≤ (3x)/(8)+O ((√(x))/(log x)). We also show that the orbit changes parity exactly at square states. On dyadic orbit segments, we establish a local-to-global criterion, a large-value truncation, and a quantitative implication from small relative variance to a step-mass-saturating dynamic near-ladder. Finally, under two explicit hypotheses -- a regularity-or-ladder dichotomy and an anti-ladder estimate -- we obtain a(x)\asymp x/log x.

Citations

• Multiplicative functions in short intervals

Related

• On an asymptotic behavior of the divisor function τ(n)
• Sums Associated with Orbits in the Binary Dynamical System
• The sequence of middle divisors is unbounded
• On Congruences for Iterates of the Sum--Power Divisor Function and Conditional Implications for the Riemann Hypothesis
• Primitive prime divisors in the critical orbit of z^d+c