Transcendental Numbers

This is an excellent book which can be used for a one- or two-semester upper undergraduate course or first or second year graduate course in transcendental numbers. The prerequisites are modest: Complex variables and a basic algebra course covering extension fields.

The book has everything you would want from a book: i) coverage of basic topics, ii) coverage of advanced topics, iii) short digestible chapters, iv) adequate exercises, v) a lean and lively look, vi) extensive bibliography, vii) sense of humor, viii) emerging elusive topics, ix) tested in the classroom.

Coverage of basic topics: For example, Chapters 1,2,3,5, and 6 cover Liouville’s, Hermite’s, and Lindemann’s Theorems, the Maximum Modulus Principle, and Siegel’s Lemma (bounds on solutions of systems of simultaneous linear equations), respectively.

Coverage of advanced topics: These include transcendental values of the following: elliptic functions, the j-function, modular forms, class group L-functions, and elliptic integrals. Transcendental results on both individual numbers as well as independence are presented. Baker’s theorem, the Baker-Birch-Wirsing theorem and applications are also included.

In passing, one interesting topic not covered in the book are several results in recent years on the finiteness of intersection of certain sets. For example, the set of numbers represented by only one digit and the set of Fibonacci numbers has finite intersection (there are only finitely many Fibonacci numbers that can be represented in base 10 with one digit, such as 55). The book also does not mention Mateev’s contributions in this area. However, this is not a serious defect, since these results are very computational and can be studied after reading the book. I suspect that the authors omitted very computational results since it would be incongruent with the readability and brief but comprehensive coverage of book topics.

Short digestible chapters: There are 28 chapters in 205 pages resulting in an average of 7 pages per chapter. Yet each of these chapters covers a major technique, a major historical development or a major advanced topic. The proofs are complete and the presentation style is almost-narrative despite the technicality of the subject.

Adequate exercises: The exercises both review and challenge. The exercises are illustrative of what the student will know after reading the book. Here are some examples

1. Use Euler’s result that the sum of the reciprocals of integer squares is \( \pi^2/6\) to show that there are an infinite number of primes,
2. Show that the sum of the reciprocals of a degree-2 or higher polynomial over all integers is 0 or transcendental (provided certain modest conditions hold on the polynomial),
3. Assuming Nesterenko’s conjecture, show the value of a non-zero modular form with algebraic Fourier coefficients evaluated at an algebraic number is transcendental (provided certain modest conditions hold on the algebraic number),
4. Show that the exponential function is algebraically independent of the Weierstrass zeta function,
5. Show that the reciprocal of a Liouville number is also a Liouville number.

A Lean and Lively look: 205 pages covering 28 chapters, each one giving complete sets of definitions and proofs for a variety of topics spanning historical and advanced transcendence concepts.

Extensive bibliography: The bibliography contains 138 references including both books and papers. The interested reader can therefore pursue in more depth individual topics.

Sense of humor: For example, “We observe this amusing corollary of the above theorem: Corollary: All of the following numbers — \(\Gamma(1/8), \Gamma(3/8),\Gamma(5/8), \Gamma(7/8)\) — are transcendental with at most one exception.”

Emerging elusive topics: The last chapter, which is self-contained, is intended to give the reader an introduction to the emerging theory of periods and multiple zeta-values.

Tested in the classroom: This book grew out of lectures given by both authors at their respective institutions. The 28 chapters can conveniently be covered in two semesters with the first semester covering basics and some advanced topics and the second semester covering more advanced topics.

Russell Jay Hendel, [email protected], holds a Ph.D. in theoretical mathematics and an Associateship from the Society of Actuaries. He teaches at Towson University. His interests include discrete number theory, applications of technology to education, problem writing, actuarial science and the interaction between mathematics, art and poetry.