The variance of integers without small prime factors in short intervals

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 $y$.