The variance of integers without small prime factors in short intervals

TL;DR

This paper investigates the variance of integers lacking small prime factors within short intervals, revealing connections to smooth numbers and providing unconditional asymptotic results that improve understanding of their distribution.

Contribution

It offers the first unconditional asymptotic analysis of this variance in short intervals, linking it to smooth number statistics and surpassing naive probabilistic models.

Findings

Variance is asymptotically smaller than naive predictions for intervals of certain lengths.

Unconditional asymptotic results are established in a specific range of interval lengths.

The study connects the distribution of such integers with properties of y-smooth numbers.

Abstract

The variance of primes in short intervals relates to the Riemann Hypothesis, Montgomery's Pair Correlation Conjecture and the Hardy--Littlewood Conjecture. In regards to its asymptotics, very little is known unconditionally. We study the variance of integers without prime factors below $y$, in short intervals. We use complex analysis and sieve theory to prove an unconditional asymptotic result in a range for which we give evidence is qualitatively best possible. We find that this variance connects with statistics of $y$-smooth numbers, and, as with primes, is asymptotically smaller than the naive probabilistic prediction once the length of the interval is at least a power of $y$.