Topics in analytic number theory Open Access

Abstract

In this thesis, the author proves results using the circle method, sieve theory and the distribution of primes, character sums, modular forms and Maass forms, and the Granville-Soundararajan theory of pretentiousness. In particular, he proves theorems about partitions and q-series, almost-prime values of polynomials, Gauss sums, modular forms, quadratic forms, and multiplicative functions exhibiting extreme cancellation. This includes a proof of the Alder-Andrews conjecture, generalizations of theorems of Iwaniec and Ono and Soundararajan, and answers to questions of Zagier and Serre, as well as questions of the author in the Granville-Soundararajan theory of pretentiousness.

Table of Contents

1 Introduction

1.1 Gauss sums 1.2 Sieve theory and the distribution of primes 1.3 The analytic theory of modular forms 1.3.1 The Alder-Andrews and Andrews conjectures 1.3.2 Eta-quotients and theta functions 1.3.3 Representation by ternary quadratic forms 1.4 The pretentious view of analytic number theory 2 Gauss sums over finite fields and roots of unity 2.1 The Gross-Koblitz formula 2.2 Proof of Theorem 2.1 3 Almost-primes represented by quadratic polynomials 3.1 Proof of Theorem 3.2 3.1.1 A weighted sum 3.1.2 The linear sieve 3.1.3 Proof of Theorem 3.2 3.2 An equidistribution result for the congruence G(x) ≡ 0 (mod m) 4 The analytic theory of modular forms 4.1 The Alder-Andrews conjecture 4.1.1 Estimate of Qd(n) with explicit error bound 4.1.2 Estimate of qd(n) with explicit error bound 4.1.3 Proof of Alder's Conjecture 4.2 A conjecture of Andrews 4.2.1 Proof of Andrews's conjecture in the limit 4.3 Eta-quotients and theta functions 4.3.1 Preliminary Facts 4.3.2 Proof of Theorem 4.13 4.4 Representation by ternary quadratic forms 4.4.1 Representation by ternary quadratic forms 4.4.2 Siegel zeros: Proof of Theorem 4.20 4.4.3 Tate-Shafarevich groups: Proof of Theorem 4.22 5 The pretentious view of analytic number theory 5.1 Multiplicative functions dictated by Artin symbols 5.1.1 Proof of Theorem 5.3 5.2 Pretentiously detecting power cancellation 5.2.1 Strong pretentiousness 5.2.2 β-pretentiousness Bibliography

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School	Laney Graduate School
Department	Math and Computer Science
Degree	PhD
Submission	Dissertation
Language	English
Research Field	Mathematics
Keyword	Analytic Number Theory
Committee Chair / Thesis Advisor	Ono, Ken, Emory University
Committee Members	Borthwick, David, Emory University Zureick-Brown, David M, Emory University

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