Sieving the positive integers by small primes

Abstract

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