# On Mahler's transcendence measure for $e$

## TL;DR

This paper provides an explicit transcendence measure for e and its powers, improving previous results by employing Hermite-Padé approximations and detailed analysis of common factors.

## Contribution

It introduces a new explicit transcendence measure for e and its powers, advancing the work of Mahler, Borel, and Hata.

## Key findings

- Explicit transcendence measure for e established
- Transcendence measure for positive integer powers of e proved
- Improved bounds based on Hermite-Padé approximations

## 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.