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| Preface to the Dover Edition | |
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| Preface | |
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| Notation | |
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| Errata | |
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| Introduction | |
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| Hypotheses H and H<sub>N</sub> | |
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| Sieve methods | |
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| Scope and presentation | |
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| The Sieve of Eratosthenes: Formulation of the General Sieve | |
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| Introductory remarks | |
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| The sequences A | |
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| Basic examples | |
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| The sifting set B and the sifting function S | |
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| The sieve of Eratosthenes-Legendre | |
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| The Combinatorial Sieve | |
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| The general method | |
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| Brun's pure sieve | |
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| Technical preparation | |
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| Brun's sieve | |
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| A general upper bound O-result | |
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| Sifting by a thin set of primes | |
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| Further applications | |
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| Fundamental Lemma | |
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| Rosser's sieve | |
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| The Simplest Selberg Upper Bound Method | |
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| The method | |
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| The case �(d) = 1, R<sub>d</sub> ≤ 1 | |
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| Application to 1 | |
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| The Brun-Titchmarsh inequality | |
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| The Titchmarsh divisor problem | |
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| The case �(p) = p/p-1 | |
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| The prime twins and Goldbach problems: explicit upper bounds | |
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| The problem ap + b = p': an explicit upper bound | |
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| The Selberg Upper Bound Method (continued): O-results | |
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| A lower bound for G(x, z) | |
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| Applications | |
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| The Selberg Upper Bound Method: Explicit Estimates | |
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| A two-sided �<sub>2</sub>-condition | |
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| Technical preparation | |
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| Asymptotic formula for G(z) | |
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| The main theorems | |
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| Two ways of dealing with polynomial sequences {F(p)}: discussion | |
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| Primes representable by polynomials | |
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| Primes representable by polynomials F(p): the non-linearized approach | |
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| Prime k-tuplets | |
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| Primes representable by polynomials F(p): the linearized approach | |
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| An Extension of Selberg's Upper Bound Method | |
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| The method | |
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| An upper estimate | |
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| The function �<sub>�</sub> | |
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| Asymptotic formula for G(�, z) | |
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| The main result | |
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| Selberg's Sieve Method (continued): A First Lower Bound | |
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| Combinatorial identities | |
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| Ah asymptotic formula for S | |
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| Fundamental Lemma | |
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| The function �<sub>�</sub> | |
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| A lower bound | |
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| The main result | |
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| TheLinear Sieve | |
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| The method | |
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| The functions F, f | |
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| An approximate identity for the leading terms | |
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| Upper and lower bounds for S | |
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| The main result | |
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| A Weighted Sieve: The Linear Case | |
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| The method | |
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| Application to the prime twins and Goldbach problems | |
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| The weighted sieve in applicable form | |
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| Almost-primes in intervals and arithmetic progressions | |
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| Almost-primes representable by irreducible polynomials F(n) | |
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| Almost-primes representable by irreducible polynomials F(p) | |
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| Weighted Sieves: The General Case | |
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| The first method | |
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| The first method in applicable form | |
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| Almost-primes representable by polynomials | |
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| The second method | |
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| Almost-primes representable by polynomials | |
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| Another method | |
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| Chen's Theorem | |
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| Introduction | |
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| The weighted sieve | |
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| Application of Selberg's upper sieve | |
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| Transition to primitive characters | |
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| Application of contour integration | |
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| Application of the large sieve | |
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| Bibliography | |
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| References | |