Cover -- Contents -- Preface -- Notation -- Chapter 1. Introduction -- 1.1 The Beginning -- 1.2 The Sieve of Eratosthenes -- 1.3 The Sieve of Eratosthenes-Legendre -- 1.4 The Prime Number Theorem and Its Consequences -- 1.5 Brun, Selberg, and Rosser-Iwaniec -- 1.6 Eratosthenes-Legendre-Vinogradov -- Chapter 2. The Vaughan Identity -- 2.1 Introduction -- 2.2 An Exponential Sum over Primes -- 2.3 The Distribution of αp Modulo 1 -- 2.4 The Bombieri-Vinogradov Theorem -- 2.5 Linnik's and Heath-Brown's Identities -- 2.6 Further Thoughts on Vaughan's Identity -- Chapter 3. The Alternative Sieve -- 3.1 Introduction -- 3.2 Cosmetic Surgery -- 3.3 The Fundamental Theorem -- 3.4 Application to the Distribution of {αp} -- 3.5 A Lower-Bound Sieve -- 3.6 A Change of Notation -- 3.7 The Piatetski-Shapiro PNT -- 3.8 Historical Note -- Chapter 4. The Rosser-Iwaniec Sieve -- 4.1 Introduction -- 4.2 A Fundamental Lemma -- 4.3 A Heuristic Argument -- 4.4 Proof of the Lower-Bound Sieve -- 4.5 Developments of the Rosser-Iwaniec Sieve -- Chapter 5. Developing the Alternative Sieve -- 5.1 Introduction -- 5.2 New Forms of the Fundamental Theorem -- 5.3 Reversing Roles -- 5.4 A New Idea -- 5.5 Higher-Dimensional Versions -- 5.6 Greatest Prime Factors -- Chapter 6. An Upper-Bound Sieve -- 6.1 The Method Described -- 6.2 A Device by Chebychev -- 6.3 The Arithmetical Information -- 6.4 Applying the Rosser-Iwaniec Sieve -- 6.5 An Asymptotic Formula -- 6.6 The Alternative Sieve Applied -- 6.7 Upper-Bounds: Region by Region -- 6.8 Why a Previous Idea Fails -- Chapter 7. Primes in Short Intervals -- 7.1 The Zero-Density Approach -- 7.2 Preliminary Results -- 7.3 The 7/12 Result -- 7.4 Shorter Intervals -- 7.5 Application of Watt's Theorem -- 7.6 Sieve Asymptotic Formulae -- 7.7 The Two-Dimensional Sieve Revisited -- 7.8 Further Asymptotic Formulae -- 7.9 The Final Decomposition.

7.10 Where to Now? -- Chapter 8. The Brun-Titchmarsh Theorem on Average -- 8.1 Introduction -- 8.2 The Arithmetical Information -- 8.3 The Alternative Sieve Applied -- 8.4 The Alternative Sieve for τ ≤ α1 ≤ 3/7, θ ≤ 11/21 -- 8.5 The Alternative Sieve in Two Dimensions -- 8.6 The Alternative Sieve in Three Dimensions -- 8.7 An Upper Bound for Large θ -- 8.8 Completion of Proof -- Chapter 9. Primes in Almost All Intervals -- 9.1 Introduction -- 9.2 The Arithmetical Information -- 9.3 The Alternative Sieve Applied -- 9.4 The Final Decomposition -- 9.5 An Upper-Bound Result -- 9.6 Other Measures of Gaps Between Primes -- Chapter 10. Combination with the Vector Sieve -- 10.1 Introduction -- 10.2 Goldbach Numbers in Short Intervals -- 10.3 Proof of Theorem 10.2 -- 10.4 Dirichlet Polynomials -- 10.5 Sieving the Interval B[sub(1)] -- 10.6 Sieving the Interval B[sub(2)] -- 10.7 Further Applications -- Chapter 11. Generalizing to Algebraic Number Fields -- 11.1 Introduction -- 11.2 Gaussian Primes in Sectors -- 11.3 Notation and Outline of the Method -- 11.4 The Arithmetical Information -- 11.5 Asymptotic Formulae for Problem 1 -- 11.6 The Final Decomposition for Problem 1 -- 11.7 Prime Ideals in Small Regions -- 11.8 First Steps -- 11.9 Estimates for Dirichlet Polynomials -- 11.10 Asymptotic Formulae for Problem 2 -- 11.11 The Final Decomposition for Problem 2 -- Chapter 12. Variations on Gaussian Primes -- 12.1 Introduction -- 12.2 Outline of the Fouvry-Iwaniec Method -- 12.3 Some Preliminary Results -- 12.4 Fouvry-Iwaniec Type I Information -- 12.5 Reducing the Bilinear Form Problem -- 12.6 Catching the Cancellation Introduced by µ -- 12.7 The Main Term for Theorem 12.1 -- 12.8 The Friedlander-Iwaniec Outline for a[sup(2)] + b[sup(4)] -- 12.9 The Friedlander-Iwaniec Asymptotic Sieve -- 12.10 Sketch of the Crucial Result -- 12.11 And Now?.

Chapter 13. Primes of the Form x[sup(3)] + 2y[sup(3)] -- 13.1 Introduction -- 13.2 Outline of the Proof -- 13.3 Preliminary Results -- 13.4 The Type I Estimates -- 13.5 The Fundamental Lemma Result -- 13.6 Proof of Lemma 13.6 -- 13.7 Proof of Lemma 13.7 -- 13.8 The Type II Information Established -- Chapter 14. Epilogue -- 14.1 A Summary -- 14.2 A Challenge with Which to Close -- Appendix -- A.1 Perron's formula -- A.2 Buchstab's Function ω(u) -- A.3 Large-Sieve Inequalities -- A.4 The Mean Value Theorem for Dirichlet Polynomials -- A.5 Smooth Functions -- Bibliography -- Index -- A -- B -- C -- D -- E -- F -- G -- H -- I -- J -- K -- L -- M -- P -- R -- S -- T -- V -- W -- Y -- Z.