6
$\begingroup$

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:

  1. 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 $g(p)=1/p$?

  1. 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?
$\endgroup$

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.