Topics in analytic number theory Open Access
Thorner, Jesse (2016)
Published
Abstract
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 RankinSelberg Lfunctions. 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 logfree zero density estimates for RankinSelberg Lfunctions with effective dependence on the key parameters. From this, the author proves an approximate short interval prime number theorem for RankinSelberg Lfunctions, an approximate short interval version of the SatoTate conjecture, and a bound on the least norm of a prime ideal counted by the SatoTate 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 Lfunctions. 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 BombieriVinogradov 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 logfree zero density estimates for automorphic Lfunctions and the SatoTate 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 SatoTate conjecture. 83
4.3.3 Proof of Theorem 4.5. 85
4.4 Sym nminorants. 86
Bibliography. 88
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