I am trying to apply Iwaniec's formulation of Rosser's sieve (here) to obtain nontrivial lower bounds for almost-primes in various sequences. These sequences have sieve dimension 1 (if $g(p)$ is the fraction divisible by $p$, then $g(p) ~ 1/p + c/p^2$ for a constant $c$). It seems that to obtain good upper bounds explicit constants are unnecessary but for lower bounds I am missing two things:
- In addition to the sieve dimension, a constant $K$ is needed essentially bounding the variation at small primes:
$\displaystyle \prod_{w\le p < z} (1 - g(p))^{-1} < \frac{\log z}{\log w} (1+\frac{K}{\log w}).$
What is the optimal $K$ in the above inequality, e.g. if $g(p)=1/p$?
- What are the optimal (or best known) constants in the function $Q(s)$ in Rosser's sieve defined in Theorem 1 of the Iwaniec paper?
