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Intro; Foreword; References; Preface; Contents; Chains of Large Gaps Between Primes; 1 Introduction; 1.1 Notational Conventions; 2 Siegel Zeroes; 3 Sieving an Interval; 4 Sieving a Set of Primes; 5 Using a Hypergraph Covering Theorem; 6 Using a Sieve Weight; References; A Note on the Distribution of Primes in Intervals; 1 Introduction; 2 Some Literature; 2.1 The Singular Series on Average; 2.2 Unconditional Results; 3 Notation; 4 Ancillary Results; 4.1 Inclusion-Exclusion; 4.2 The Singular Series, As a Series; 4.3 The Singular Series, on Average; 5 Main Results; 6 Concluding Remarks.
On the Difference in Values of the Euler Totient FunctionNear Prime Arguments1 Introduction; 2 Proof of Theorem 1 for = 1; 2.1 The Case of Equality; 2.2 A Comparison Lemma; 2.3 Some Preliminaries; 2.4 Inconvenient Primes of Type 1; 2.5 Inconvenient Primes of Type 2; 2.6 Inconvenient Primes of Type 3; 2.7 Inconvenient Primes of Type 4; 2.8 Convenient Primes; 2.9 Weird Primes; 2.10 Conclusion; 2.11 Sanity Check; 3 Proof of Theorem 1 for 2; 3.1 The Case of Equality; 3.2 A More General Comparison Lemma; 3.3 Final Ingredient; References.
Vinogradov's Mean Value Theorem as an Ingredient in Polynomial Large Sieve Inequalities and Some Consequences1 Introduction; 2 Overview of Current Polynomial LSI Bounds and Conjectures; 3 The Path from VMVT to Polynomial Large Sieve Inequalities; 4 Fraction Counting: Bounds for the Spacing Problem; 5 Consequences of the Polynomial LSI for the Distribution of Primes in Certain Arithmetic Progressions; References; Unexpected Regularities in the Behavior of Some Number-Theoretic Power Series; 1 Introduction; 2 Fake Asymptotics and the Fröberg Heuristic; 3 Better than Expected Error Terms.
4 Fake Asymptotics for Other Number-Theoretic Power Series4.1 Möbius Series with ``Scaled'' Coefficients; 4.2 Möbius Series with Non-principal Characters; 4.3 Möbius Series with Coprimality Condition; 4.4 The Liouville Power Series; 5 Concluding Remarks; References; The Convex Hull of the Prime Number Graph; 1 Introduction; 2 Counting the Convex Primes; 3 Edge Convex Primes; 4 Multiplicatively Convex Primes; 5 Data and Future Work; References; Irregular Behaviour of Class Numbers and Euler-Kronecker Constants of Cyclotomic Fields: The Log Log Log Devil at Play; 1 Introduction.
On the Difference in Values of the Euler Totient FunctionNear Prime Arguments1 Introduction; 2 Proof of Theorem 1 for = 1; 2.1 The Case of Equality; 2.2 A Comparison Lemma; 2.3 Some Preliminaries; 2.4 Inconvenient Primes of Type 1; 2.5 Inconvenient Primes of Type 2; 2.6 Inconvenient Primes of Type 3; 2.7 Inconvenient Primes of Type 4; 2.8 Convenient Primes; 2.9 Weird Primes; 2.10 Conclusion; 2.11 Sanity Check; 3 Proof of Theorem 1 for 2; 3.1 The Case of Equality; 3.2 A More General Comparison Lemma; 3.3 Final Ingredient; References.
Vinogradov's Mean Value Theorem as an Ingredient in Polynomial Large Sieve Inequalities and Some Consequences1 Introduction; 2 Overview of Current Polynomial LSI Bounds and Conjectures; 3 The Path from VMVT to Polynomial Large Sieve Inequalities; 4 Fraction Counting: Bounds for the Spacing Problem; 5 Consequences of the Polynomial LSI for the Distribution of Primes in Certain Arithmetic Progressions; References; Unexpected Regularities in the Behavior of Some Number-Theoretic Power Series; 1 Introduction; 2 Fake Asymptotics and the Fröberg Heuristic; 3 Better than Expected Error Terms.
4 Fake Asymptotics for Other Number-Theoretic Power Series4.1 Möbius Series with ``Scaled'' Coefficients; 4.2 Möbius Series with Non-principal Characters; 4.3 Möbius Series with Coprimality Condition; 4.4 The Liouville Power Series; 5 Concluding Remarks; References; The Convex Hull of the Prime Number Graph; 1 Introduction; 2 Counting the Convex Primes; 3 Edge Convex Primes; 4 Multiplicatively Convex Primes; 5 Data and Future Work; References; Irregular Behaviour of Class Numbers and Euler-Kronecker Constants of Cyclotomic Fields: The Log Log Log Devil at Play; 1 Introduction.