Bounded gaps between primes in Chebotarev sets

Jesse Thorner

doi:10.1186/2197-9847-1-4

Abstract

A new and exciting breakthrough due to Maynard establishes that there exist infinitely many pairs of distinct primes $p_1,p_2$ with $|p_1-p_2|\leq 600$ as a consequence of the Bombieri-Vinogradov Theorem. In this paper, we apply his general method to the setting of Chebotarev sets of primes. We study applications of these bounded gaps with an emphasis on ranks of prime quadratic twists of elliptic curves over $\mathbb{Q}$, congruence properties of the Fourier coefficients of normalized Hecke eigenforms, and representations of primes by binary quadratic forms.

Highlights

Introduction and statement of results The long-standing twin prime conjecture states that there are infinitely many primes p such that p + 2 is prime
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
The fact that there is a large amount of numerical evidence supporting the twin prime conjecture is fascinating, considering that the Prime Number Theorem tells us that on average, the gap between consecutive primes p1, p2 is about log(p1)

Summary

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

Background

Methods

Results and discussions

Conclusion