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

  • Kurt Mahler

Abstract

In Part I, Mahler introduces his classification of complex numbers and the following two results are proved. Let ϑ1,ϑ2,,ϑN\vartheta_1,\vartheta_2,\ldots,\vartheta_N be NN algebraic numbers that are linearly independent over the rationals and let λ\lambda be a Liouville number. Then, the numbers eϑ1,eϑ2,,eϑN,λe^{\vartheta_1},e^{\vartheta_2},\ldots,e^{\vartheta_N},\lambda are algebraically independent over the field of algebraic numbers. Let zz be the real logarithm of a positive rational number not equal to one and let λ\lambda be a Liouville number. Then, zz and λ\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 π\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)].