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Exceptional characters and prime numbers in sparse sets

Jori Merikoski
Vol. 18 (2024), No. 7, 1305–1332
DOI: 10.2140/ant.2024.18.1305
Abstract

We develop a lower bound sieve for primes under the (unlikely) assumption of infinitely many exceptional characters. Compared with the illusory sieve due to Friedlander and Iwaniec which produces asymptotic formulas, we show that less arithmetic information is required to prove nontrivial lower bounds. As an application of our method, assuming the existence of infinitely many exceptional characters we show that there are infinitely many primes of the form a2 + b8.

Keywords
prime numbers, Siegel zero, sieve methods, prime values of polynomials, exceptional character
Mathematical Subject Classification
Primary: 11N32
Secondary: 11N36
Milestones
Received: 31 August 2022
Revised: 19 June 2023
Accepted: 3 September 2023
Published: 13 June 2024
Authors
Jori Merikoski
Mathematical Institute
University of Oxford
Oxford
United Kingdom
© 2024 MSP (Mathematical Sciences Publishers). Distributed under the Creative Commons Attribution License 4.0 (CC BY).

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