The Hurwitz Zeta Function and the Lerch Zeta Function

  • Markus Szymon Fraczek
Part of the Lecture Notes in Mathematics book series (LNM, volume 2139)


In this chapter we will discuss formulas we have developed for the evaluation of certain zeta functions. We will need them later for the numerical computation of the spectrum of the transfer operator. The implementations of these zeta functions are in a sense the heart of our computations, so we need to be very careful. Unfortunately we have found approximations of these zeta functions in the literature which are quite limited; this is the case especially for the Lerch zeta function. This is understandable, since these zeta functions are special functions, which compared to others are not used so often.


  1. 2.
    Abramowitz, M., Stegun, I.: Handbook of Mathematical Functions. Dover, New York (1964)zbMATHGoogle Scholar
  2. 27.
    Cartier, P.: An introduction to zeta functions, chap. 1 In: Waldschmidt, M., Moussa, P., Luck, J.-M., ItzyksonSpringer, C. (eds.) From Number Theroy to Phyiscs. Springer, Berlin/New York (1992)Google Scholar
  3. 39.
    Erdelyi, A.: Higher Transcendental Functions, vol. 1. McGraw-Hill, New York (1953)zbMATHGoogle Scholar
  4. 65.
    Hurwitz, A.: Einige Eigenschaften der Dirichlet’schen Funktionen \(F(s) =\sum \frac{D} {n} \cdot \frac{1} {n^{s}}\), die bei der Bestimmung der Klassenanzahlen Binärer quadratischer Formen auftreten. Z. für Math. und Physik 27, 86–101 (1882)Google Scholar
  5. 77.
    Lerch, M.: Note sur la fonction K(w, x, s) =  n ≥ 1exp{2π i n x}(n + w)s. Acta Math. 11, 19–24 (1887)Google Scholar
  6. 98.
    Mühlenbruch, T.: Hecke operators on period functions for the full modular group. Ph.D. thesis, Cuvillier Verlag (2003)Google Scholar

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© Springer International Publishing AG 2017

Authors and Affiliations

  • Markus Szymon Fraczek
    • 1
  1. 1.Mathematics InstituteUniversity of WarwickCoventryUK

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