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| Preface | |
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| List of notation | |
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| Dirichlet series: I | |
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| Generating functions and asymptotics | |
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| Analytic properties of Dirichlet series | |
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| Euler products and the zeta function | |
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| Notes | |
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| References | |
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| The elementary theory of arithmetic functions | |
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| Mean values | |
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| The prime number estimates of Chebyshev and of Mertens | |
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| Applications to arithmetic functions | |
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| The distribution of [Omega](n) - [omega](n) | |
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| Notes | |
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| References | |
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| Principles and first examples of sieve methods | |
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| Initiation | |
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| The Selberg lambda-squared method | |
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| Sifting an arithmetic progression | |
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| Twin primes | |
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| Notes | |
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| References | |
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| Primes in arithmetic progressions: I | |
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| Additive characters | |
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| Dirichlet characters | |
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| Dirichlet L-functions | |
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| Notes | |
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| References | |
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| Dirichlet series: II | |
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| The inverse Mellin transform | |
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| Summability | |
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| Notes | |
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| References | |
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| The Prime Number Theorem | |
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| A zero-free region | |
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| The Prime Number Theorem | |
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| Notes | |
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| References | |
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| Applications of the Prime Number Theorem | |
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| Numbers composed of small primes | |
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| Numbers composed of large primes | |
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| Primes in short intervals | |
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| Numbers composed of a prescribed number of primes | |
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| Notes | |
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| References | |
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| Further discussion of the Prime Number Theorem | |
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| Relations equivalent to the Prime Number Theorem | |
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| An elementary proof of the Prime Number Theorem | |
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| The Wiener-Ikehara Tauberian theorem | |
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| Beurling's generalized prime numbers | |
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| Notes | |
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| References | |
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| Primitive characters and Gauss sums | |
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| Primitive characters | |
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| Gauss sums | |
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| Quadratic characters | |
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| Incomplete character sums | |
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| Notes | |
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| References | |
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| Analytic properties of the zeta function and L-functions | |
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| Functional equations and analytic continuation | |
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| Products and sums over zeros | |
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| Notes | |
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| References | |
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| Primes in arithmetic progressions: II | |
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| A zero-free region | |
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| Exceptional zeros | |
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| The Prime Number Theorem for arithmetic progressions | |
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| Applications | |
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| Notes | |
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| References | |
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| Explicit formulae | |
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| Classical formulae | |
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| Weil's explicit formula | |
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| Notes | |
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| References | |
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| Conditional estimates | |
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| Estimates for primes | |
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| Estimates for the zeta function | |
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| Notes | |
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| References | |
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| Zeros | |
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| General distribution of the zeros | |
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| Zeros on the critical line | |
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| Notes | |
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| References | |
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| Oscillations of error terms | |
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| Applications of Landau's theorem | |
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| The error term in the Prime Number Theorem | |
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| Notes | |
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| References | |
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| Appendices | |
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| A The Riemann-Stieltjes integral | |
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| Notes | |
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| References | |
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| Bernoulli numbers and the Euler-MacLaurin summation formula | |
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| Notes | |
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| References | |
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| The gamma function | |
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| Notes | |
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| References | |
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| Topics in harmonic analysis | |
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| Pointwise convergence of Fourier series | |
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| The Poisson summation formula | |
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| Notes | |
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| References | |
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| Name index | |
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| Subject index | |