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
5 - Integral functions of finite order
Published online by Cambridge University Press: 05 August 2012
- 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 analytic number-theory almost all the analytic functions which are used satisfy certain growth conditions at infinity. These can be subsumed in the general condition that the functions are ‘integral of finite order’. In particular this class of functions can be represented by infinite products (as we used in §3.1) and the norms describing the growth satisfy various convexity properties (as we used in §2.12, for more details see Appendix 6). In this appendix we shall summarise those aspects of the theory of integral functions of finite order which are useful for analytic number-theory. For the sake of completeness we shall also sketch proofs.
The basic theorem in all investigations of this type is Jensen's Theorem, which is stated below. It is in the same cadre as Cauchy's Theorem but is a ‘real variable’ theorem and it is this fact that accounts for its usefulness.
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- An Introduction to the Theory of the Riemann Zeta-Function , pp. 138 - 145Publisher: Cambridge University PressPrint publication year: 1988