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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
a 2
+ b 8 .
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
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