| Preface | p. xi |
| Notations, Cross-references, References | p. xiii |
| Historical introduction | p. 1 |
| Definition of the zeta-function | |
| Euler's proof of the infinity of primes | |
| The values of [zeta](2k) | |
| Riemann and the zeta-function | |
| The Hadamard Product Formula | |
| The explicit formula | |
| The Prime Number Theorem | |
| The Riemann Hypothesis and the Prime Number Theorem | |
| The approximate functional equation | |
| Disclaimer | |
| Some number-theoretic functions | |
| Exercises | p. 10 |
| The Poisson Summation Formula and the functional equation | p. 14 |
| Summation formulae | |
| Poisson Summation Formula | |
| Mellin transform | |
| Cut-off functions | |
| Integral representation | |
| Local functional equation | |
| Analytic continuation and the functional equation | |
| Proofs | |
| Variant | |
| Further variant | |
| Estimates | |
| Growth in vertical strips | |
| First step towards the 'approximate functional equation' | |
| Derivation of the Poisson Summation Formula from properties of the zeta-function | |
| Remarks | |
| Exercises | p. 27 |
| The Hadamard Product Formula and 'explicit formulae' of prime number theory | p. 33 |
| Hadamard Product Formula | |
| Expansion of the logarithmic derivative | |
| Estimates for the logarithmic derivative | |
| Estimate on the zeros | |
| Proofs | |
| Explicit formula | |
| Proof | |
| Explicit formula, approximate version | |
| Explicit formula, classical version | |
| Proof | |
| Closing remarks | |
| Exercises | p. 46 |
| The zeros of the zeta-function and the Prime Number Theorem | p. 50 |
| Introductory remarks | |
| Hadamard's zero-free region | |
| Prime Number Theorem, first version | |
| Proof of theorem 4.3 | |
| Prime Number Theorem, second version | |
| The functions N and S | |
| The basic estimate of S | |
| Further remarks | |
| Asymptotic estimate for N | |
| Proof of Theorem 4.9 | |
| The function N([sigma], T) | |
| The quadratic mean of the zeta-function | |
| Proof of Theorem 4.11 | |
| Proof of Theorem 4.12 | |
| General remarks on higher means | |
| Exercises | p. 64 |
| The Riemann Hypothesis and the Lindelof Hypothesis | p. 67 |
| Formulation of the Riemann Hypothesis | |
| Formulation of the Lindelof Hypothesis | |
| Equivalent forms of the Lindelof Hypothesis | |
| The Lindelof Hypothesis and the zeros | |
| The Riemann Hypothesis implies the Lindelof Hypothesis | |
| Proof of Theorem 5.4 | |
| A further consequence of the Lindelof Hypothesis | |
| The Riemann Hypothesis and prime numbers | |
| Proof of Theorem 5.8 | |
| Reasons for believing the Riemann Hypothesis | |
| Zeta-functions of curves over finite fields | |
| Places of a function-field | |
| The zeta-function of a function-field | |
| Analogue of the 'explicit formulae' | |
| Weil's criterion for the Riemann Hypothesis for the zeta-function of a function-field | |
| Weil's criterion for the Riemann Hypothesis | |
| Weil's criterion and the Jacobian of the curve | |
| Significance of the previous section for the Riemann Hypothesis | |
| Concluding remarks on the Riemann Hypothesis | |
| Exercises | p. 87 |
| The approximate functional equation | p. 91 |
| Formulation of the approximate functional equation | |
| A special case | |
| Estimates of certain integrals | |
| Proof of Theorem 6.1 | |
| Integral means | |
| The estimate [zeta](1/2 + it) = O( | |
| Van der Corput's Summation Formula | |
| Weyl's Lemma | |
| Exponential sums | |
| More estimates on exponential sums | |
| Proof of Theorem 6.6 | |
| Exercises | p. 108 |
| Appendices | |
| Fourier theory | p. 111 |
| The Riemann-Lebesgue Lemma | |
| The variation of a function | |
| A lemma | |
| The Dirichlet kernel | |
| Representation of a periodic function by a Fourier series | |
| A useful estimate | |
| The L[superscript 2]-theory | |
| Fourier integrals and the inversion formula | |
| Another useful estimate | |
| The Plancherel Theorem | |
| Convolution of periodic functions | |
| Convolution of integrable functions | |
| The Poisson Summation Formula | |
| Exercises | p. 119 |
| The Mellin transform | p. 122 |
| Definition of the Mellin transform | |
| The inversion formula | |
| Convolution | |
| A variant | |
| Perron's Formula | |
| Exercises | p. 124 |
| An estimate for certain integrals | p. 127 |
| An estimate for certain integrals | |
| Complements | |
| Further complements | |
| Exercises | p. 128 |
| The gamma-function | p. 130 |
| Definition of the gamma-function | |
| The beta-function | |
| The infinite product | |
| The logarithmic derivative | |
| A further integral formula | |
| Stirling's Theorem | |
| The functional equation | |
| Stirling's Theorem. | |
| Exercises | p. 135 |
| Integral functions of finite order | p. 138 |
| Introductory remarks | |
| Jensen's Theorem | |
| Integral functions of finite order | |
| Integral functions of finite order without poles or zeros | |
| The general representation theorem | |
| A Maximum Principle for integral functions | |
| An extension of the previous section | |
| The Phragmen-Lindelof Theorem | |
| A consequence of the Phragmen-Lindelof Theorem | |
| Other convexity theorems | |
| Exercises | p. 145 |
| Borel-Caratheodory Theorems | p. 146 |
| General remarks | |
| The Borel-Caratheodory Theorem | |
| An estimate of the argument | |
| Littlewood's Theorem | p. 148 |
| Littlewood's Theorem | |
| A theorem of Backlund | |
| Bibliography | p. 152 |
| Index | p. 155 |
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