Jesse Thorner

Bounded gaps between primes in Chebotarev sets

AbstractPurposeA new and exciting breakthrough due to Maynard establishes that there exist infinitely many pairs of distinct primes p1, p2 with |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.MethodsWe use recent developments in sieve theory due to Maynard and Tao in conjunction with standard results in algebraic number theory.ResultsGiven a Galois extension K/Q, we prove the existence of bounded gaps between primes p having the same Artin symbol K/Qp.ConclusionsWe 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 classificationPrimary 11N05; 11N36; Secondary 11G05

The explicit Sato-Tate Conjecture and densities pertaining to Lehmer-type questions

Let

A Chebotarev Variant of the Brun–Titchmarsh Theorem and Bounds for the Lang-Trotter conjectures

f(z)=\sum_{n=1}^\infty a(n)q^n\in S^{\text{new}}_ k (\Gamma_0(N))

The error term in the Sato–Tate conjecture

be a newform with squarefree level

Benford’s Law for coefficients of newforms

N

Special values of motivic L -functions and zeta-polynomials for symmetric powers of elliptic curves

that does not have complex multiplication. For a prime

Effective Log-Free Zero Density Estimates for Automorphic L-Functions and the Sato–Tate Conjecture

p

An explicit bound for the least prime ideal in the Chebotarev density theorem

, define

Bounded Gaps Between Primes in Multidimensional Hecke Equidistribution Problems

\theta_p\in[0,\pi]

A unified and improved Chebotarev density theorem

to be the angle for which