PDF files of articles are only available for institutions which have paid for the online version upon signing an Institutional User License.

Sieving intervals and Siegel zeros

Volume 205 / 2022

Andrew Granville Acta Arithmetica 205 (2022), 1-19 MSC: Primary 11M20; Secondary 11N35. DOI: 10.4064/aa201002-25-6 Published online: 5 September 2022


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.


  • Andrew GranvilleDépartment de Mathématiques et Statistique
    Université de Montréal
    CP 6128 succ. Centre-Ville
    Montréal, QC H3C 3J7, Canada

Search for IMPAN publications

Query phrase too short. Type at least 4 characters.

Rewrite code from the image

Reload image

Reload image