## 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