Unpublished result of Rosser in Sieve Methods book

Question

Erdős and Selfridge (1971) state that the following is "implied by an unpublished result of Rosser" which they claim appears in a forthcoming book on sieve methods by Halberstam and Richert. I guess I could try to find something implying it in that 400-page book (called "Sieve Methods", available online in unsearchable format) but I thought I'd ask if maybe somebody knows a precise reference [or a proof!] for this result?

For every $\epsilon>0$, there is a constant $c=c(\epsilon)$ such that at least $c\frac{k}{\log k}$ integers between $n$ and $n+k$ have all their prime factors greater than $k^{1/2-\epsilon}$.

Answer 1

This is part of the theory of the "linear sieve." Chapter 8 of the book of Halberstam and Richert deals with this topic. Alternatively you could look at Iwaniec's paper On the error term in the linear sieve. The result you want can be extracted from the lower bound (1.4) on page 2 of Iwaniec's paper, together with the formula for $f(s)$ at the top of that page.