An effective Bombieri-Vinogradov error term for sifting problems
Daniel R. Johnston
TL;DR
The paper addresses the ineffectivity inherent in Bombieri–Vinogradov-type sieve bounds by developing an effective variant that preserves the original asymptotics. It achieves this by reformulating sieve upper and lower bounds to circumvent Siegel zeros, leveraging BV-type bounds in primes in arithmetic progressions, and yielding fully computable constants. As key outcomes, it proves effective upper and lower bounds and applies them to twin primes and Goldbach-type representations with an explicit constant of $(4+\varepsilon)$, along with a two-dimensional sieve application giving representations of integers as a prime-quadratic residue combination with a bounded prime-factor term. These results enhance the practical applicability of BV-based sieve methods, enabling explicit, computable bounds in central problems of additive prime number theory.
Abstract
In number theory, many major results related to the twin prime and Goldbach conjectures 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*} is the twin-prime constant.