Textbook in PDF format This long-anticipated work shares the aims of its celebrated companion: namely, to provide an introduction for students and a reference for researchers to the techniques, results, and terminology of multiplicative number theory. This volume builds on the earlier one (which served as an introduction to basic, classical results) and focuses on sieve methods. This area has witnessed a number of major advances in recent years, e.g. gaps between primes, large values of Dirichlet polynomials and zero density estimates, all of which feature here. Despite the fact that the book can serve as an entry to contemporary mathematics, it remains largely self-contained, with appendices containing background or material more advanced than undergraduate mathematics. Preface Notation Exponential Sums I: Van der Corput’s Method Exponential Integrals Elementary Estimates Van der Corput’s Method Notes References Estimates for Sums over Primes Principles of the Method An Exponential Sum Formed with Primes Further Applications Digit Sums of Primes Notes References Additive Prime Number Theory Sums of Three Primes Sums of Two Primes on Average Conditional Estimates A Lower Bound for the Error Term The Distribution of Primes in Short Intervals Notes References The Large Sieve Trigonometric Polynomials Mean Square Distribution in Arithmetic Progressions Character Sums Maximal Variants Notes References Primes in Arithmetic Progressions: III ........................ The Bombieri–Vinogradov Theorem Applications of the Bombieri–Vinogradov Theorem Mean Square Distribution Notes References Sieves of Fixed Dimension and Applications Refresher on Sieves The Rosser–Iwaniec Sieve The Linear Sieve The Selberg Examples Some Applications of Sieve Theory Almost Primes in Polynomial Sequences Notes References Bounded Gaps between Primes The GPY Sieve The Proof of Maynard’s Theorem Consequences of Maynard’s Theorem Notes References Appendix E Topics in Harmonic Analysis II Uniform Approximation of Continuous Functions Quantitative Trigonometric Approximation An Additional Trigonometric Majorant Maximal Inequalities Notes References Appendix F Uniform Distribution Uniform Distribution (mod 1) Quantitative Estimates Kronecker’s Theorem Almost Periodicity Notes References Appendix G Bounds for Bilinear Forms The Operator Norm of a Matrix Square Matrices Bessel’s Inequality Hilbert’s Inequality Notes References Appendix H Linear Programming Fundamental Theory The Application to Sieves Notes References Errata for Volume 1 Name Index Subject Index