Consecutive Almost-Primes

For ϵ>0 we define
P(ϵ)={n∈N: n=pk, p prime, k≤n}.
Thus if ϵ is small, the elements of P(ϵ) are “almost prime”. (Note that this phrase is used here in a different sense from that of the weighted sieve, in which n=P_r is an almost-prime if n has at most r prime factors.) It was conjectured by Erdos [1] that for any ϵ>0 the set P(ϵ) contains infinitely many consecutive pairs n, n+1. This was proved recently by Hildebrand [4]. In the present paper we shall investigate quantitative forms of Hildebrand’s result.