Topics in analytic number theory Open Access

Thorner, Jesse (2016)

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In this thesis, the author proves theorems on the distribution of primes by extending recent results in sieve theory and proving new results on the distribution of zeros of Rankin-Selberg L-functions. The author proves for any number field K which is a Galois extension of the rational numbers, there exist bounded gaps between primes with a given "splitting condition" in K, and the primes in question may be restricted to short intervals. Furthermore, we can count these gaps with the correct order of magnitude. The author also proves log-free zero density estimates for Rankin-Selberg L-functions with effective dependence on the key parameters. From this, the author proves an approximate short interval prime number theorem for Rankin-Selberg L-functions, an approximate short interval version of the Sato-Tate conjecture, and a bound on the least norm of a prime ideal counted by the Sato-Tate conjecture, all of which exhibit effective dependence on the key parameters.

Table of Contents

1 Introduction. 1

1.1 The distribution of primes. 1

1.2 Gaps between primes. 3

1.3 The distribution of primes in short intervals. 6

1.4 The distribution of zeros of L-functions. 10

2 Bounded gaps between primes in Chebotarev sets. 13

2.1 Notation. 14

2.2 Bounded gaps between primes. 14

2.3 Proof of Theorem 2.1. 17

2.4 Applications to number fields and elliptic curves. 27

2.5 Applications to modular forms and quadratic forms. 34

3 A variant of the Bombieri-Vinogradov theorem for short intervals and some questions of Serre. 36

3.1 Preliminary Setup. 39

3.2 Proof of Theorem 3.1. 41

3.3 Bounded gaps between primes in Chebotarev sets: the short interval version. 43

3.4 Arithmetic applications. 47

4 Effective log-free zero density estimates for automorphic L-functions and the Sato-Tate conjecture. 51

4.1 Preliminaries. 63

4.1.1 Definitions and notation. 63

4.1.2 Preliminary lemmata. 67

4.2 The zero density estimate. 71

4.2.1 Proof o fTheorems 4.2 and 4.3. 72

4.2.2 Bounds on derivatives. 73

4.2.3 Zero detection: The proof of Proposition 4.11. 78

4.3 Arithmetic Consequences. 79

4.3.1 Setup and proof of Theorem 4.4. 79

4.3.2 The Sato-Tate conjecture. 83

4.3.3 Proof of Theorem 4.5. 85

4.4 Sym n-minorants. 86

Bibliography. 88

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