Bounded gaps between primes in Chebotarev sets

Abstract

Purpose: A new and exciting breakthrough due to Maynard establishes that there exist infinitely many pairs of distinct primes p1, p2with |p1− p2| ≤ 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 (Formula presented). Conclusions: We study applications of these bounded gaps with an emphasis on ranks of prime quadratic twists of elliptic curves over ℚ, congruence properties of the Fourier coefficients of normalized Hecke eigenforms, and representations of primes by binary quadratic forms.