After reading Introduction to Analytic Number Theory one is left with the impression that the author, Tom M. Apostol, has pulled off some magic trick. In less than 340 pages he has covered much more material than he could possibly have covered in one book. This is even more surprising when, at least after one first quick reading, one feels that the pace is good, the proofs are detailed, and the material is nicely motivated throughout with interesting historical remarks.
Number theory is typically subdivided in three subareas: the very basics are covered in what we call “elementary number theory”, and advanced topics fall under either “algebraic number theory” or “analytic number theory”, depending on the type of statements (of theorems and conjectures) and the methods employed in the proofs. Even though the book under review is self-titled an introduction to analytic number theory, the text briefly reviews most of the basic topics that we consider “elementary number theory”: divisibility, prime numbers, the fundamental theorem of arithmetic, modular arithmetic, systems of congruences, quadratic residues, the law of quadratic reciprocity and primitive roots. But, of course, the text covers much more than elementary topics.
The book begins with a beautiful introduction to number theory and in the first few chapters the author develops the theory of arithmetical functions (i.e., the Möbius, Euler totient and Von Mangoldt functions), Dirichlet multiplication of arithmetical functions, the exact and average order of growth, identities for partial sums of Dirichlet products, etc. In Chapter 4, this theory is put to good use in order to prove several elementary theorems on the distribution of prime numbers. In particular, the author establishes several equivalent formulations of the prime number theorem, and shows theorems such as Shapiro’s tauberian theorem or Selberg’s asymptotic formula. As an application of Selberg’s formula, the author includes here a brief sketch of an “elementary” proof of the prime number theorem due to Selberg and Erdös. The full complex-analytic proof of the primer number theorem is delayed until Chapter 13.
After a brief summary of the main results in modular arithmetic, Chapter 6 is dedicated to finite abelian groups and their Dirichlet characters (the first few sections form a very brief introduction to group theory). This chapter covers enough preliminaries for the proof, in the next chapter, of one of the jewels of analytic number theory (and one of the jewels of this book): Dirichlet’s theorem on primes in arithmetic progressions. The proof offered in the book is the simplified version of Dirichlet’s original proof that was found by Harold Shapiro in 1950.
The book continues with a chapter on periodic arithmetical functions and Gauss sums, and somewhat standard chapters on Gauss’ quadratic reciprocity law, and another one on primitive roots. After this interlude of elementary number theory, the author goes into the study of zeta functions. First, in Chapter 11, he covers the basic analytic properties of Dirichlet series and their Euler products. The following chapter develops further properties of the Riemann zeta function, Dirichlet L-functions, the gamma function and the Hurwitz zeta function, including discussions about their analytic continuation, functional equations and critical values in terms of Bernoulli numbers. The exposition culminates in Chapter 13 with yet another jewel of number theory: an analytic proof of the prime number theorem based on the analytic properties of the Riemann zeta function (the strategy is to express Chebyshev’s function in terms of the Riemann zeta function by means of a contour integral).
As if all this was not enough, there is one last chapter on partitions that includes several interesting results: a proof of Euler’s pentagonal-number theorem, Euler’s recursion formula, Jacobi’s triple product identity and several partition identities due to Ramanujan.
Perhaps it is a good time for a disclaimer: I am a number theorist, therefore highly biased, but I must admit that I love this book. The selection of topics is excellent, the exposition is fluid, the proofs are clear and nicely structured, and every chapter contains its own set of (20 to 40) exercises. But, obviously, I am not an undergraduate learning analytic number theory for the first time. However, I can say that I would have loved to read this book when I was a student. The chapters on elementary number theory are a nice addition in order to make this book self-contained, but are too brief to replace an introductory course on the fundamentals of arithmetic. Analytic number theory is a tough subject, that tends to be dry and technical, but this book is very readable and approachable, and it would work very nicely as a text for a second course in number theory.
But wait, wait! There is even more! Apostol’s graduate textbook Modular functions and Dirichlet Series in Number Theory is a second volume to the book under review, which I will review separately. So look for my review of the second part elsewhere in this site.
Álvaro Lozano-Robledo is Assistant Professor of Mathematics and Associate Director of the Q Center at the University of Connecticut. He is the author of Elliptic Curves, Modular Forms, and Their L-functions.