Topics in analytic number theory Open Access
Lemke Oliver, Robert (2013)
Published
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 GranvilleSoundararajan theory of pretentiousness. In particular, he proves theorems about partitions and qseries, almostprime values of polynomials, Gauss sums, modular forms, quadratic forms, and multiplicative functions exhibiting extreme cancellation. This includes a proof of the AlderAndrews 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 GranvilleSoundararajan 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 AlderAndrews and Andrews conjectures 1.3.2 Etaquotients 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 GrossKoblitz formula 2.2 Proof of Theorem 2.1 3 Almostprimes 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 AlderAndrews 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 Etaquotients 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 TateShafarevich 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 βpretentiousness BibliographyAbout this Dissertation
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