Analytic number theory is a discipline that uses the tools of analysis — primarily complex analysis and Fourier series — to study the properties of the integers. At most universities, it is typically taught as a graduate course with complex analysis as a prerequisite. Stopple has challenged this orthodoxy by writing a textbook in analytic number theory that is designed for an undergraduate course and that does not assume a background in complex variables. As the subtitle indicates, he has also included considerable material on the history of number theory.
The book starts with a discussion of polygonal numbers; this motivates a discussion of finite differences, harmonic series, and geometric series. The second chapter covers sums of divisors and perfect numbers. A quote from Nicomachus about the distribution of perfect numbers motivates the discussion of orders of magnitude in the third chapter, and leads to the introduction of the notations "O" and "<<". Both of these symbols are introduced in a section entitled "Landau notation." This is slightly misleading, for while the "O" notation is traditionally attributed to Landau, the "<<" is usually attributed to Vinogradov.
The distribution of primes is a central subject in analytic number theory, and this topic is introduced in Chapter 5 with a heuristic discussion of the prime number theorem. This is followed up with proofs of the Chebyshev bounds for primes. Chapter 6 discusses Euler's formulas for ζ(2k), and this leads into a discussion of the Euler product formula for the Riemann zeta-function in Chapter 7. After a brief review of complex numbers, the author discusses the analytic continuation and functional equation of the zeta-function in Chapters 8 and 9. The Hadamard product formula for ξ(s) is given without proof in Chapter 10; with this as a starting point, the author develops the explicit formulas of von Mangoldt and Riemann.
Chapters 11-13 are brief introductions to other topics. In Chapter 11, Stopple discusses Pell's equation and introduces Dirichlet L-functions. Chapter 12 covers elliptic curves, and it includes expository accounts of Wiles' proof of Fermat's Last Theorem and of the Birch-Swinnerton-Dyer conjecture. Chapter 13 is entitled "Analytic Theory of Algebraic Numbers;" it includes a discussion of binary quadratic forms, of the class number formula for negative discriminants, and of Siegel zeros.
The greatest advantage of this book is that it walks the reader through a lot of details that other texts take for granted. For example, when the "O" notation is first used, Stopple is very careful to indicate all of the details of the relevant inequalities. Moreover, there are several chapters entitled "Interludes" that give review of important background material. The first interlude is on calculus, and Stopple introduces it by saying "It covers things I wish you had learned but, based on my experience, I expect you did not." (Most faculty will recognize the sentiment.) Another feature is that there is material that is not easily available from other sources; e.g., the material on finite differences in the first chapter.
The historical part of the book consists of well-placed brief vignettes of people who have contributed to the subject. In doing this, Stopple had to achieve a delicate balance — too little detail would devolve into a sterile list of names and dates, while too much detail would become too great a distraction. His approach is to treat each person with one or two paragraphs that include some revealing anecdote. I am not an expert in the history of mathematics, and I cannot vouch for the accuracy of his accounts, but the vignettes are interesting and well done. They add to the pleasure of reading the book.
Of course, there are drawbacks to not requiring complex analysis. There is nothing on zero-free regions for ζ(s), and the prime number theorem is never proved. Nonetheless, the book accomplishes its aim of providing an introduction to analytic number theory that is accessible to undergraduates. The graduate student planning to specialize in number theory will want to study books such as Apostol or Davenport that require complex analysis. But such a student could also profitably use this book as supplementary reading, particularly when (s)he is trying to puzzle out the intricacies of "O".
T. Apostol, Introduction to Analytic Number Theory, fifth edition. Undergraduate Texts in Mathematics, Springer-Verlag, New York, 1995.
H. Davenport, Multiplicative Number Theory, third edition, revised and with a preface by H. L. Montgomery. Graduate Texts in Mathematics, vol. 74, Springer-Verlag, New York, 2000.
S. W. Graham is currently a Professor of Mathematics at Central Michigan University. He can be reached at firstname.lastname@example.org.
1. Sums and differences; 2. Products and divisibility; 3. Order and magnitude; 4. Counterexamples; 5. Averages; 6. Prime number theorems; 7. Series; 8. The Basel problem; 9. Euler’s product; 10. The Riemann zeta function; 11. Pell’s equation; 12. Elliptic curves; 13. Symmetry; 14. Explicit formula.