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Extension to Other Zeta- and L-Functions

  • André VorosEmail author
Chapter
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Part of the Lecture Notes of the Unione Matematica Italiana book series (UMILN, volume 8)

Abstract

In this chapter, based on [107], we extend the treatment and results of the three former chapters to the setting where the Riemann zeros are replaced by the (nontrivial) zeros of a more general zeta or L-function L(x), still fairly similar to ζ(x). We will use the terms: primary function for this function L(x) which supplies the new zeros in this extended setting, and Riemann case for the former setting where ζ(x) itself was the fixed primary function.

The interest of this extension is twofold. First, it broadens the previous results in a natural way: with little work, we will accommodate three distinct kinds of superzeta functions as before, but now over the (nontrivial) zeros of numerous primary functions. Second, it sheds some further light on the results for ζ(x) itself: the origin of many final values in the previous chapters will be clarified through their more abstract specifications. For instance, various special values like \(\mathop C\nolimits_\alpha ^{{\rm IER}}\) will now explicitly stem from the Stirling expansion (10.12) for the trivial factor \({\mathbf G} (x), x \rightarrow + \infty.\).

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Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  1. 1.CEA-Saclay Institut de Physique Théorique (IPhT) Orme des MerisiersGif-sur-YvetteFrance

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