Queen's University

Transcendental Numbers

This book provides an introduction to the topic of transcendental numbers for upper-level undergraduate and graduate students. The text is constructed to support a full course on the subject, including descriptions of both relevant theorems and their applications. While the first part of the book focuses on introducing key concepts, the second part presents more complex material, including applications of Baker’s theorem, Schanuel’s conjecture, and Schneider’s theorem. These later chapters may be of interest to researchers interested in examining the relationship between transcendence and L-functions. Readers of this text should possess basic knowledge of complex analysis and elementary algebraic number theory.

Contents

Liouville’s Theorem 1
Hermite’s Theorem 7
Lindemann’s Theorem 11
The Lindemann–Weierstrass Theorem 15
The Maximum Modulus Principle and Its Applications 19
Siegel’s Lemma 23
The Six Exponentials Theorem 27
Estimates for Derivatives 31
The Schneider–Lang Theorem 35
Elliptic Functions 39
Transcendental Values of Elliptic Functions 49
Periods and Quasiperiods 55
Transcendental Values of Some Elliptic Integrals 59
The Modular Invariant 65
Transcendental Values of the j-Function 75
More Elliptic Integrals 79
Transcendental Values of Eisenstein Series 83
Elliptic Integrals and Hypergeometric Series 89
Baker’s Theorem 95
Some Applications of Baker’s Theorem 101
Schanuel’s Conjecture 111
Transcendental Values of Some Dirichlet Series 123
The Baker–Birch–Wirsing Theorem 131
Transcendence of Some Infinite Series 137
Linear Independence of Values of Dirichlet L-Functions 153
Transcendence of Values of Class Group L-Functions 159
Transcendence of Values of Modular Forms 179
Periods, Multiple Zeta Functions and ζ(3) 185