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Published online by Cambridge University Press: 05 August 2012
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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