Book contents
- Frontmatter
- Contents
- Preface
- Notations, Cross-references, References
- 1 Historical introduction
- 2 The Poisson Summation Formula and the functional equation
- 3 The Hadamard Product Formula and ‘explicit formulae’ of prime number theory
- 4 The zeros of the zeta-function and the Prime Number Theorem
- 5 The Riemann Hypothesis and the Lindelöf Hypothesis
- 6 The approximate functional equation
- Appendices
- 1 Fourier theory
- 2 The Mellin transform
- 3 An estimate for certain integrals
- 4 The gamma-function
- 5 Integral functions of finite order
- 6 Borel–Caratheodory Theorems
- 7 Littlewood's Theorem
- Bibliography
- Index
6 - Borel–Caratheodory Theorems
Published online by Cambridge University Press: 05 August 2012
Book contents
- Frontmatter
- Contents
- Preface
- Notations, Cross-references, References
- 1 Historical introduction
- 2 The Poisson Summation Formula and the functional equation
- 3 The Hadamard Product Formula and ‘explicit formulae’ of prime number theory
- 4 The zeros of the zeta-function and the Prime Number Theorem
- 5 The Riemann Hypothesis and the Lindelöf Hypothesis
- 6 The approximate functional equation
- Appendices
- 1 Fourier theory
- 2 The Mellin transform
- 3 An estimate for certain integrals
- 4 The gamma-function
- 5 Integral functions of finite order
- 6 Borel–Caratheodory Theorems
- 7 Littlewood's Theorem
- Bibliography
- Index
Summary
In the main text of this book the use of such theorems as we shall briefly describe here has been avoided. For more sophisticated applications this is not possible and so for the sake of completeness two relatively simple results of this type will be given which are often useful. This type of theorem is closely related to the Maximum Principle but characteristic of the class of Borel–Caratheodory Theorems is that one assumes only a bound on the real part and deduces one on the imaginary part from this.
- Type
- Chapter
- Information
- An Introduction to the Theory of the Riemann Zeta-Function , pp. 146 - 147Publisher: Cambridge University PressPrint publication year: 1988