Bounded gaps between primes in Chebotarev sets

• Published in: Research in the mathematical sciences Vol. 1; no. 1
• Main Author: Thorner, Jesse
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01.12.2014
• Subjects: Applications of Mathematics; Computational Mathematics and Numerical Analysis; Mathematics; Mathematics and Statistics; Chebotarev Density Theorem; Twin primes; Elliptic curves
• ISSN: 2197-9847; 2197-9847
• DOI: 10.1186/2197-9847-1-4

Purpose A new and exciting breakthrough due to Maynard establishes that there exist infinitely many pairs of distinct primes p 1 , p 2 with | p 1 - p 2 | ≤ 600 as a consequence of the Bombieri-Vinogradov Theorem. In this paper, we apply his general method to the setting of Chebotarev sets of primes. Methods We use recent developments in sieve theory due to Maynard and Tao in conjunction with standard results in algebraic number theory. Results Given a Galois extension K / Q , we prove the existence of bounded gaps between primes p having the same Artin symbol K / Q p . Conclusions We study applications of these bounded gaps with an emphasis on ranks of prime quadratic twists of elliptic curves over , congruence properties of the Fourier coefficients of normalized Hecke eigenforms, and representations of primes by binary quadratic forms. AMS subject classification Primary 11N05; 11N36; Secondary 11G05