Sieving intervals and Siegel zeros
Volume 205 / 2022
Acta Arithmetica 205 (2022), 1-19
MSC: Primary 11M20; Secondary 11N35.
DOI: 10.4064/aa201002-25-6
Published online: 5 September 2022
Abstract
Assuming that there exist (infinitely many) Siegel zeros, we show that the (Rosser–)Jurkat–Richert bounds in the linear sieve cannot be improved, and similarly look at Iwaniec’s lower bound on Jacobsthal’s problem, as well as minor improvements to the Brun–Titchmarsh Theorem. We also develop a suggestion by Ford to rework Cramér’s heuristic to show that we would expect gaps around $x$ that are significantly larger than $(\log x)^2$ if there are infinitely many Siegel zeros.
