Complex Tauberian Theorems

  • Jacob Korevaar
Part of the Grundlehren der mathematischen Wissenschaften book series (GL, volume 329)


In the course of Chapters I and II, the notion of ‘Tauberian theorem’ has evolved. We would now say that such a theorem involves a class of objects S (functions, series, sequences) and a transformation T. The transformation is an ‘averaging operation’ with attendant continuity property: certain limit behavior of the original S implies related limit behavior of the image ’T S. The aim of a Tauberian theorem is to reverse the averaging, or pass to a different average. One wants to go from a limit property of T S to a limit property of S, or another transform of S. Such theorems typically require an additional condition, a ‘Tauberian condition’, on S and perhaps a condition on the transform T S.


Zeta Function Dirichlet Series Boundary Behavior Tauberian Theorem Prime Number Theorem 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Jacob Korevaar
    • 1
  1. 1.KdV Mathematical InstituteUniversity of AmsterdamAmsterdamThe Netherlands

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