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 = 1/2 / 4.3.3: |
A Sieve Theorem Following Rosser / 4.4: |
The Case k >/= 1/2: a First Result / 4.4.1: |
Theorem 1 when k= 1/2 / 4.4.2: |
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: |