Mahler, Kurt

Reprint: Arithmetische Eigenschaften der Lösungen einer Klasse von Funktionalgleichungen (1929)

Doc. Math. Extra Vol. Mahler Selecta, 191-216 (2019)
DOI: 10.25537/10.25537/dm.2019.SB-191-216


In this paper, Mahler introduces a method of transcendence now known as Mahler's method. He applies his method to show that the numbers \par \[ \par \par \sum_{n\ge 0} t(n)\alpha^n\quad\mbox{and}\quad\sum_{n\ge 0}\lfloor n\omega\rfloor \alpha^n \par \par \] \par are transcendental for any algebraic number \(\alpha\) with \(0<|\alpha|<1\) and any positive quadratic irrational number \(\omega\), where \(\{t(n)\}_{n\ge 0}\) is the Thue-Morse sequence with values in \(\{-1,1\}\) and \(\rfloor x\lfloor\) denotes the integer part of \(x\). \par This article is the first in a series of three papers that develops Mahler's method. \par Reprint of the author's paper [Math. Ann. 101, 342--366 (1929; JFM 55.0115.01); correction 103, 532 (1930; JFM 56.0185.02)].

Mathematics Subject Classification

11-03, 11J81


Mahler's method