An effective Bombieri-Vinogradov error term for sifting problems

Abstract

In number theory, many major results related to the additive properties of primes are proven using the methods of sieve theory. However, in nearly every case, the existing proofs of these results are ineffective, in that explicit values for which they hold cannot be computed. The reason for this ineffectivity is due to the reliance on the Bombieri--Vinogradov theorem. In this paper, we show that any classical sifting problem with a Bombieri--Vinogradov style error term can in fact be made effective, with no loss to the asymptotic form of the original (ineffective) result. This is done by carefully modifying the sieve upper and lower bounds as to avoid the usual complications regarding the existence of a Siegel zero. We also provide some simple applications. For example, we show that one may effectively bound the number of primes $p\leq x$ such that $p+2$ is also prime by \begin{equation*}(4+o(1))C_2\frac{x}{(\log x)^2},\end{equation*}where\begin{equation*}C_2=2\prod_{p>2}\left(1-\frac{1}{(p-1)^2}\right).\end{equation*}