On Mahler's transcendence measure for $e$

Abstract

We present a completely explicit transcendence measure for $e$. This is a continuation and an improvement to the works of Borel, Mahler and Hata on the topic. Furthermore, we also prove a transcendence measure for an arbitrary positive integer power of $e$. The results are based on Hermite-Pad\'e approximations and on careful analysis of common factors in the footsteps of Hata.