Short intervals with a given number of primes

A well-known conjecture asserts that, for any given positive real number λ and nonnegative integer m  , the proportion of positive integers n⩽xn⩽x for which the interval (n,n+λlog⁡n](n,n+λlog⁡n] contains exactly m   primes is asymptotically equal to λme−λ/m!λme−λ/m! as x tends to infinity. We show that the number of such n   is at least x1−o(1)x1−o(1).