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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 | |