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Guided by Schwarz’ Functions: A Walk Through the Garden of Mahler’s Transcendence Method

In: From Arithmetic to Zeta-Functions

Author

Listed:
  • Peter Bundschuh

    (Universität zu Köln, Mathematisches Institut)

  • Keijo Väänänen

    (University of Oulu, Department of Mathematical Sciences)

Abstract

In this paper, transcendence results and, more generally, results on the algebraic independence of functions and their values are proved via Mahler’s analytic method. Here the key point is that the functions involved satisfy certain types of functional equations as G d (z d ) = G d (z) − z∕(1 − z) in the case of $$G_{d}(z):=\sum _{h\geq 0}z^{d^{h} }/(1 - z^{d^{h} })$$ for d ∈ { 2, 3, 4, …}. In 1967, these particular functions G d (z) were arithmetically studied by W. Schwarz using Thue–Siegel–Roth’s approximation method.

Suggested Citation

  • Peter Bundschuh & Keijo Väänänen, 2016. "Guided by Schwarz’ Functions: A Walk Through the Garden of Mahler’s Transcendence Method," Springer Books, in: Jürgen Sander & Jörn Steuding & Rasa Steuding (ed.), From Arithmetic to Zeta-Functions, pages 91-101, Springer.
    • Handle: RePEc:spr:sprchp:978-3-319-28203-9_6
    DOI: 10.1007/978-3-319-28203-9_6
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