A sharp refresh scale for power-free interval counts
Year: 2026
Type: preprint
Abstract: The random-sieve model of power-free integers assigns an independent uniform forbidden residue class modulo $p^k$ to each prime $p$. We determine how the centered count in an interval of length $H$ responds when each prime coordinate is independently resampled with probability $\varepsilon_H$. For every fixed integer $k\ge2$, the correlation between the original and resampled counts tends to $e^{-t}$ when $\varepsilon_H\log\log H\to t\in[0,\infty)$, and tends to zero when this product diverges. Thus $1/\log\log H$ is the sharp refresh scale. The proof expresses the covariance as a positive arithmetic sum and transfers the classical normal order of the number of distinct prime factors to its bounded weights. An exact cancellation of Euler factors then gives the correlation profile. As a consequence, the number of prime coordinates in a character sampled according to the interval count's normalized Fourier energy, divided by $\log\log H$, converges in probability to one.
Author: Vaseghi, Sam
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225558209