No abstract

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 Q, congruence properties of the Fourier coefficients of normalized Hecke eigenforms, and representations of primes by binary quadratic forms.

In [29], the first author obtained a weak subconvexity result bounding central values of a large class of L-functions, assuming a weak Ramanujan hypothesis on the size of Dirichlet series coefficients of the L-function. If C denotes the analytic conductor of the L-function in question, then C 1 4 is the size of the convexity bound, and the weak subconvexity bound established there was of the form C 1 4 /(log C) 1−ǫ . In this paper we establish a weak subconvexity bound of the shape C 1 4 /(log C) δ for some small δ > 0, but with a much milder hypothesis on the size of the Dirichlet series coefficients. In particular our results will apply to all automorphic L-functions, and (with mild restrictions) to the Rankin-Selberg L-functions attached to two automorphic representations.In order to make clear the scope and limitations of our results, we axiomatize the properties of L-functions that we need. In Section 2 we shall discuss how automorphic L-functions and Rankin-Selberg L-functions fit into this framework. Let m ≥ 1 be a natural number. We now describe axiomatically a class of L-functions, which we shall denote by S(m).1. Dirichlet series and Euler product. The functions L(s, π) appearing in the class S(m) will be given by a Dirichlet series and Euler productwith both the series and the product converging absolutely for Re(s) > 1. It will also be convenient for us to writeSetting λ π (n) = 0 if n is not a prime power, we haveλ π (n)Λ(n) n s , and log L(s, π) = ∞ n=2 λ π (n)Λ(n) n s log n . Functional equation.Writewhere N π ≥ 1 is known as the "conductor" of the L-function and the µ π (j) are complex numbers. We suppose that there is an integer 0 ≤ r = r π ≤ m such that the completed Date: October 31, 2018. 1 2 KANNAN SOUNDARARAJAN AND JESSE THORNERL-function s r (1 − s) r L(s, π)L ∞ (s, π) extends to an entire function of order 1, and satisifies the functional equationHere κ π is a complex number with |κ π | = 1, andWe suppose that r has been chosen such that the completed L-function does not vanish at s = 1 and s = 0. Thus, if L(s, π) has a pole at s = 1 then we are assuming that the order of this pole is at most m, and r is taken to be the order of the pole. If L(s, π) has no pole at s = 1, then we take r = 0 and are making the assumption that the L(1, π) = 0. In our work, a key measure of the "complexity" of the L-function L(s, π) is the "analytic conductor" which is defined to be (1.7) C(π) = N π m j=1

We improve the Chebotarev variant of the Brun-Titchmarsh theorem proven by Lagarias, Montgomery, and Odlyzko using the log-free zero density estimate and zero repulsion phenomenon for Hecke L-functions that were recently proved by the authors. Our result produces an improvement for the best unconditional bounds toward two conjectures of Lang and Trotter regarding the distribution of traces of Frobenius for elliptic curves and holomorphic cuspidal modular forms. We also obtain new results on the distribution of primes represented by positive-definite integral binary quadratic forms.2 Theorem 5.1 of [20] actually claims a stronger result, but a step in the proof seems not to be justified. The best that the argument appears to give is what we have stated above; see Section 9 for details.3 Note that we recover Murty's claimed result [20, Theorem 5.1].