| Introduction |
| The Structure of Sifting Arguments / 1： |
| The Sieves of Eratosthenes and Legendre / 1.1： |
| The Contribution of Eratosthenes / 1.1.1： |
| Legendre's Sieve / 1.1.2： |
| An Estimate for n(X) / 1.1.3： |
| The Distribution of Primes / 1.1.4： |
| Examples of Sifting Situations / 1.2： |
| Notations / 1.2.1： |
| The Integers in an Interval (Y - X, Y ) / 1.2.2： |
| Numbers Given by Polynomial Expressions / 1.2.3： |
| Arithmetic Progressions / 1.2.4： |
| Sums of Two Squares / 1.2.5： |
| Polynomials with Prime Arguments / 1.2.6： |
| A General Formulation of a Sifting Situation / 1.3： |
| The Basic Formulation / 1.3.1： |
| Legendre's Sieve in a General Setting / 1.3.2： |
| A Generalised Formulation / 1.3.3： |
| A Further Generalisation / 1.3.4： |
| Sifting Density / 1.3.5： |
| The Sifting Limit Β(k) / 1.3.6： |
| Composition of Sieves / 1.3.7： |
| Notes on Chapter 1 / 1.4： |
| Selberg's Upper Bound Method / 2： |
| The Sifting Apparatus / 2.1： |
| Selberg's Theorem / 2.1.1： |
| The Numbers (lambda)(d) / 2.1.2： |
| A Simple Application / 2.1.3： |
| General Estimates of G(x) and E(D, P) / 2.2： |
| An Estimate by Rankin's Device / 2.2.1： |
| Asymptotic Formulas / 2.2.2： |
| The Error Term / 2.2.3： |
| Applications / 2.3： |
| Prime Twins and Goldbach's Problem / 2.3.1： |
| Polynomial Sequences / 2.3.3： |
| Notes on Chapter 2 / 2.4： |
| Combinatorial Methods / 3： |
| The Construction of Combinatorial Sieves / 3.1： |
| Preliminary Discussion of Brun's Ideas / 3.1.1： |
| Fundamental Inequalities and Identities / 3.1.2： |
| Buchstab's Identity / 3.1.3： |
| The Combinatorial Sieve Lemma / 3.1.4： |
| Brun's Pure Sieve / 3.2： |
| Inequalities and Identities / 3.2.1： |
| The "Pure Sieve" Theorem / 3.2.2： |
| A Corollary / 3.2.3： |
| Prime Twins / 3.2.4： |
| A Modern Edition of Brun's Sieve / 3.3： |
| Rosser's Choice of X / 3.3.1： |
| A Technical Estimate / 3.3.2： |
| A Simplifying Approximation / 3.3.3： |
| A Combinatorial Sieve Theorem / 3.3.4： |
| Brun's Version of his Method / 3.3.5： |
| Brun's Choice of x / 3.4.1： |
| The Estimations / 3.4.2： |
| The Result / 3.4.3： |
| Notes on Chapter 3 / 3.5： |
| Rosser's Sieve / 4： |
| Approximations by Continuous Functions / 4.1： |
| The Recurrence Relations / 4.1.1： |
| Partial Summation / 4.1.2： |
| The Leading Terms / 4.1.3： |
| The Functions F and f / 4.2： |
| The Difference-Differential Equations / 4.2.1： |
| The Adjoint Equation and the Inner Product / 4.2.2： |
| Solutions of the Adjoint Equation / 4.2.3： |
| Particular Values of F(s) and f(s) / 4.2.4： |
| Asymptotic Analysis as k -> $(infinity$) / 4.2.5： |
| The Convergence Problem / 4.3： |
| The Auxiliary Functions / 4.3.1： |
| Adjoints and Inner Products / 4.3.2： |
| The Case k
|
| A Sieve Theorem Following Rosser / 4.4： |
| The Case k >/= 1/2: a First Result / 4.4.1： |
| Theorem 1 when k
|
| An Improved Version of Proposition 1 / 4.4.3： |
| A Two-Sided Estimate / 4.4.4： |
| Extremal Examples / 4.5： |
| The Linear Case / 4.5.1： |
| The Case k=1/2 / 4.5.2： |
| Notes on Chapter 4 / 4.6： |
| The Sieve with Weights / 5： |
| Simpler Weighting Devices / 5.1： |
| Logarithmic Weights / 5.1.1： |
| Modified Logarithmic Weights / 5.1.2： |
| Some Applications / 5.1.3： |
| More Elaborate Weighted Sieves / 5.2： |
| An Improved Weighting Device / 5.2.1： |
| Buchstab's Weights / 5.2.2： |
| A Weighted Sieve Following Rosser / 5.3： |
| Combining Sieving and Weighting / 5.3.1： |
| The Reduction Identities / 5.3.2： |
| An Identity for the Main Term / 5.3.3： |
| The Estimate for the Main Term / 5.3.4： |
| Notes on Chapter 5 / 5.4： |
| The Remainder Term in the Linear Sieve / 6： |
| The Bilinear Nature of Rosser's Construction / 6.1： |
| The Factorisation of x.d / 6.1.1： |
| Discretisations of Rosser's Sieve / 6.1.2： |
| Specification of Details / 6.1.3： |
| The Leading Contributions to the Main Term / 6.1.4： |
| The Remainder Term / 6.1.5： |
| Sifting Short Intervals / 6.2： |
| The Smoothed Formulation / 6.2.1： |
| The Remainder Sums / 6.2.2： |
| Trigonometrical Sums / 6.2.3： |
| Notes on Chapter 6 / 6.3： |
| Lower Bound Sieves when k > 1 / 7： |
| An Extension of Selberg's Upper Bound / 7.1： |
| The Integral Equation and the Function $(sigma$) (s) / 7.1.1： |
| The Estimation of G(s) / 7.1.2： |
| A Lower Bound Sieve via Buchstab's Identity / 7.2： |
| Buchstab's Iterations / 7.2.1： |
| The Buchstab Transform of the $(lambda$)2 Method / 7.2.2： |
| The Sifting Limit as k -> $(infinity$) / 7.2.3： |
| Selberg's a2 a" Method / 7.3： |
| The Improved Sifting Limit for Large k / 7.3.1： |
| Notes on Chapter 7 / 7.4： |
| References |
| Index |
| Introduction |
| The Structure of Sifting Arguments / 1： |
| The Sieves of Eratosthenes and Legendre / 1.1： |
図書
