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.

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