Reprint: On the approximation of the exponential function and the logarithm. I, II (1931/32)

Abstract

In Part I, Mahler introduces his classification of complex numbers and the following two results are proved. Let ${\vartheta }_1,{\vartheta }_2,\dots ,{\vartheta }_N$ be $N$ algebraic numbers that are linearly independent over the rationals and let $\lambda$ be a Liouville number. Then, the numbers $e^{{\vartheta }_1},e^{{\vartheta }_2},\dots ,e^{{\vartheta }_N},\lambda$ are algebraically independent over the field of algebraic numbers. Let $z$ be the real logarithm of a positive rational number not equal to one and let $\lambda$ be a Liouville number. Then, $z$ and $\lambda$ are algebraically independent over the field of algebraic numbers.

Part II continues the study of the same title by giving various bounds on polynomials evaluated at logarithms and exponentials. A new proof of the transcendence of $\pi$ is given as an application.

Reprint of the author's papers [J. Reine Angew. Math. 166, 118--136 (1931; Zbl 0003.15101; JFM 57.0242.03); ibid. 166, 137--150 (1932; Zbl 0003.38805; JFM 58.0207.01)].