Mahler's method

Abstract

Mahler's method, a term coined much later by van der Poorten, originated in three papers of K. Mahler \[Math. Ann. 101, 342--366 (1929; JFM 55.0115.01); Math. Ann. 103, 573--587 (1930; JFM 56.0185.03); Math. Z. 32, 545--585 (1930; JFM 56.0186.01)] published in 1929 and 1930. As reported in \[K. Mahler, J. Number Theory 14, 121--155 (1982; Zbl 0482.10002); A. J. van der Poorten, J. Aust. Math. Soc., Ser. A 51, No. 3, 343--380 (1991; Zbl 0738.01015), Appendix II], Mahler was really sick and laid up in bed around 1926--27 when he started to occupy himself by playing with the function $$\mathfrak{f}(z)=\sum^\infty\_{n=0} z^{2^n}.$$ While trying to show the irrationality of the number $\mathfrak{f}(p/q)$ for rational numbers $p/q$ with $0<|p/q|<1$, he finally finished proving the following much stronger statement. \[Theorem]{.smallcaps} 0.1. Let $\alpha$ be an algebraic number such that $0<|\alpha|<1$. Then $\mathfrak{f}(\alpha)$ is a transcendental number. And Mahler's method, an entirely new subject, was born. In the hands of Mahler, the method already culminated with the transcendence of various numbers such as $$\sum\_{n=0}^\infty\alpha^{2^n}, \prod\_{n=0}^\infty (1-\alpha^{2^n}), \sum\_{n=0}^\infty\lfloor n\sqrt{5}\rfloor\alpha^n, \cfrac{1}{\alpha^{-2} + \cfrac{1}{\alpha^{-4}+{\cfrac{1}{\alpha^{-8} +\cdots}}}}$$ and with the algebraic independence of the numbers $\mathfrak{f}(\alpha)$, $\mathfrak{f}'(\alpha)$, $\mathfrak{f}''(\alpha), \ldots$. Here, $\alpha$ denotes again an algebraic number with $0<|\alpha|<1$. Moreover, examples of this kind can be produced at will, as illustrated for instance in \[A. J. van der Poorten, in: Sémin. Théor. Nombres 1975--1976, Univ. Bordeaux, Exposé No. 14, 13 p. (1976; Zbl 0356.10028)]. Not only was Mahler's contribution fundamental, but also some of his ideas, described in \[K. Mahler, J. Number Theory 1, 512--521 (1969; Zbl 0184.07602)], were very influential for the future development of the theory by other mathematicians. There are several surveys including a discussion on this topic, as well as seminar reports, due to J. H. Loxton \[Bull. Aust. Math. Soc. 29, 127--136 (1984; Zbl 0519.10022); in: New advances in transcendence theory, Proc. Symp., Durham/UK 1986, 215--228 (1988; Zbl 0656.10032)], J. H. Loxton and A. J. van der Poorten \[in: Transcend. Theory, Proc. Conf., Cambridge 1976, 211--226 (1977; Zbl 0378.10020)], K. Mahler \[in: 1969 Number Theory Institute, Proc. Sympos. Pure Math. 20, 248--274 (1971; Zbl 0213.32703)], D. Masser \[Lect. Notes Math. 1819, 1--51 (2003; Zbl 1049.11081)], Yu. V. Nesterenko \[in: Proceedings of the international congress of mathematicians (ICM), August 21--29, 1990, Kyoto, Japan. Volume I. Tokyo etc.: Springer-Verlag. 447--457 (1991; Zbl 0743.11035)], K. Nishioka \[Mahler functions and transcendence. Berlin: Springer (1996; Zbl 0876.11034)], F. Pellarin \[Astérisque 317, 205--242, Exp. No. 973 (2008; Zbl 1185.11048); "An introduction to Mahler's method for transcendence and algebraic independence", Preprint, arXiv:1005.1216{.uri}], A. J. van der Poorten \[Sémin. Théor. Nombres 1974--1975, Univ. Bordeaux, Exp. No. 7, 13 p. (1975; Zbl 0331.10018); Sémin. Théor. Nombres 1975--1976, Univ. Bordeaux, Exp. No. 14, 13 p. (1976; Zbl 0356.10028); Sémin. Théor. Nombres 1986--1987, Exp. No. 27, 11 p.]. In particular, Nishioka \[loc. cit.] wrote the first and, up to date, the only book entirely devoted to Mahler's method. It provides an invaluable source of information, as well as an exhaustive account up to 1996. The author is indebted to all these mathematicians whose writings helped him a lot to prepare the present survey. He also thanks Michel Waldschmidt for his comments regarding a preliminary version of this text.