Sieving the positive integers by small primes

Abstract:

Let $Q$ be a set of primes that has relative density $\delta$ among the primes, and let $\phi (x,\,y,\,Q)$ be the number of positive integers $\leqslant x$ that have no prime factor $\leqslant y$ from the set $Q$. Standard sieve methods do not seem to give an asymptotic formula for $\phi (x,\,y,\,Q)$ in the case that $\tfrac {1}{2} \leqslant \delta < 1$. We use a method of Hildebrand to prove that \[ \phi (x,y,Q) \sim x f(u) \prod _{\substack {p < y\\p \in Q}} {\left ( {1 - \frac {1}{p}} \right )} \] as $x \to \infty$, where $u = \frac {{\log x}}{{\log y}}$ and $f(u)$ is defined by \[ {u^\delta }f(u) = \left \{ {\begin {array}{*{20}{c}} {\frac {{{e^{{\gamma ^\delta }}}}}{{\Gamma (1 - \delta )}},} \hfill & {0 < u \leqslant 1,} \hfill \\ {\frac {{{e^{{\gamma ^\delta }}}}}{{\Gamma (1 - \delta )}} + \delta \int _0^{u - 1} {f(t){{(1 + t)}^{\delta - 1}}\;dt,} } \hfill & {u > 1.} \hfill \\ \end {array} } \right .\] This may also be viewed as a generalization of work by Buchstab and de Bruijn, who considered the case where $Q$ consisted of all primes.