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The Family of the Third Kind \(\{{\mathfrak 3} (s | \tau)\}\)

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

Abstract

From Sect. 5.3, we recall the definition [51, 106]
$$\mathfrak{Z}(s | \tau) = \sum\limits_{k = 1}^{\infty}(\tau_{k} + \tau)^{-s},\qquad {\rm Re}\,\, s > 1.$$
(9.1)
For general \(\tau (\neq 0), \mathfrak{Z}(s | \tau)\) is built on a one-sided set of zeros, say \((\frac{1}{2} + {\rm i}\tau_{k})\) only; hence it seems harder to tackle than the other two families, and the earlier studies (just quoted) have revealed more singular features indeed.

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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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