Almost primes in almost all short intervals

Abstract

Let E-k be the set of positive integers having exactly k prime factors. We show that almost all intervals [x, x + log(1+epsilon) x] contain E-3 numbers, and almost all intervals [x, x + log(3.51) x] contain E-2 numbers. By this we mean that there are only 0(X) integers 1 <= x <= X for which the mentioned intervals do not contain such numbers. The result for E-3 numbers is optimal up to the epsilon in the exponent. The theorem on E-2 numbers improves a result of Harman, which had the exponent 7+epsilon in place of 3.51. We also consider general E-k numbers, and find them on intervals whose lengths approach log x as k -> infinity.