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.