*Modular Functions and Dirichlet Series in Number Theory* is, technically, the second volume of Apostol’s introduction to analytic number theory. The first volume appeared in Springer’s Undergraduate Texts in Mathematics series, and I have written a (very positive!) review for this site. Both volumes grew out of the notes for a course that Apostol had already offered at Caltech for over 25 years when the books were written.

The first volume of the two covers most of the basic topics that we consider “elementary number theory” (congruences, quadratic reciprocity, etc.) and many introductory topics from the analytic theory of numbers: the theory of arithmetical functions, several theorems on the distribution of prime numbers (including a proof of the prime number theorem), the theory of Dirichlet series and Euler products, Dirichlet’s theorem on primes in arithmetic progressions and even a chapter on partitions.

The second volume, the book under review, is mostly devoted to the theory of elliptic and modular functions, plus a chapter on diophantine approximation and a chapter on general Dirichlet series and Bohr’s equivalence. Although Apostol’s style is impeccable, and the topics that are included in this book are of classical and modern interest, the book fails to mention a number of important themes and connections that are nowadays fundamental and indispensable in any account of number theory. I will summarize the contents first, and come back to this point at the end of the review.

The first two chapters cover the basic elements about doubly periodic functions, elliptic functions, the modular group and modular functions, paying special attention to Eisenstein series, the discriminant function and Klein’s modular j-invariant function. The properties of the Dedekind eta function (e.g., its functional equation and infinite product expansion) and of Dedekind sums (e.g., reciprocity laws, congruences among different sums) are the object of study of Chapter 3.

In Chapter 4 the author describes and proves several congruences among the coefficients in the Fourier expansion of the j function. In doing so, a particular congruence subgroup of the modular group and modular forms for this group appear naturally in the discussion. This serves as motivation for a full discussion of modular forms in Chapter 6, although only modular forms for the full modular group are treated there. Before getting into the business of modular forms, in Chapter 5 the author showcases an interesting application of Dedekind’s eta function: Rademacher’s series for the partition function. The proof of Rademacher’s exact formula for the partition function is not only interesting as an application of the eta function, but also because it “represents one of the crowning achievements of the so-called circle method of Hardy, Ramanujan and Littlewood”, in Apostol’s own words.

As we already mentioned, Chapter 6 is an introduction to the rich theory of modular forms (for the full modular group, only). In this chapter the author defines modular forms, discusses the dimension formulas for the spaces of modular forms of weight k, introduces and discusses the properties of Hecke operators, and describes how one can construct an L-function from the q-expansion of a modular form. The rest of the chapter summarizes the work of Hecke which shows that certain L-functions coming from modular forms have an Euler product, an analytic continuation to the complex plane and a functional equation.

In Chapter 7, the book changes topic a bit abruptly. Here the author offers a crash-course on diophantine approximation and highlights the approximation theorems of Dirichlet, Liouville and Kronecker, together with an extension of Kronecker’s theorem to simultaneous approximation of several real numbers. Two applications of Kronecker’s theorem are discussed: (a) determine least and greatest upper bounds on the modulus of the values of the Riemann zeta function on vertical lines in the complex plane, and (b) the fact that there are no non-constant meromorphic functions with three linearly independent periods (elliptic functions are meromorphic functions that are doubly-periodic).

Kronecker’s approximation theorem is applied once again in Chapter 8, during a discussion about general Dirichlet series and the methods developed by Harald Bohr to study the set of values taken by Dirichlet series in the complex half-plane. Applications are given to the Riemann zeta-function (e.g., Turán’s theorem on some bounds that imply the Riemann hypothesis) and to Dirichlet L-functions. The second edition of this book contains an additional appendix with an alternate proof of Dedekind’s functional equation.

As I wrote in my previous review, the first book of this two-volume series is a fantastic introduction to analytic number theory for undergraduates. However, I was a little bit disappointed with this second volume since, in my opinion, certain crucial topics are missing and this renders the book somewhat outdated. Namely, in light of the celebrated modularity theorem (a.k.a. the Taniyama-Shimura-Weil conjecture) and the Birch and Swinnerton-Dyer conjecture (a.k.a., the BSD conjecture, one of the Millenium Prize Problems proposed by the Clay Math Institute, with a $1M reward), nowadays it is almost unthinkable to discuss elliptic functions and modular forms without mentioning elliptic curves (and, to a lesser degree, modular curves), and yet, elliptic curves are never mentioned in the book (not once, except for titles in the bibliography!). Just in case the reader is wondering about the timing, both the TSW and the BSD conjectures were formalized in the 1960s, and this book was written in 1976.

The strange thing is that all the tools and machinery necessary to make the connections between elliptic functions, modular forms and elliptic curves are present in the book, but the author never takes the last step across the bridge to the realm of elliptic curves. For instance, early in Chapter 1 the author proves the differential equation satisfied by the Weierstrass ℘-function which provides an isomorphism between a complex torus and an algebraic model for an elliptic curve. In fact, in Chapter 2, the author also proves (in disguise) the so-called uniformization theorem that shows that every elliptic curve is isomorphic to a complex torus. More surprisingly, the author discusses the work of Hecke at the end of Chapter 6 — which, via the modularity theorem, beautifully characterizes what modular forms correspond to elliptic curves — but this connection is never mentioned in the book. After reading the book, I get an eerie feeling that elliptic curves were meant to be discussed in this book, but the author never got the chance to write this missing chapter.

The first volume in the two-part series is exemplary in the amount of motivation and examples that are incorporated into the text. However, this second volume has little motivation, and what is there is sort of vague and self-referential. For instance, Chapter 3 begins with this sentence: “In many applications of elliptic modular functions to number theory the eta function plays a central role”. However, the author does not provide any examples. To be sure, the eta function is used in Chapter 5 to prove Rademacher’s formula — it would have been nice to point this out, at least, in Chapter 3! Another example: in Chapter 4, as an application of the theory of elliptic modular functions, the author proves several congruences among the coefficients of the Fourier expansion of the j-invariant function — but, since no other applications of the j function are given, this application is left somewhat by itself. In Chapter 6, the motivation for the theory of modular forms is also disappointing: as presented, the goal of theory is to find all the modular functions whose Fourier coefficients share the same properties as the modular discriminant and the Eisenstein series. But there are no examples in the book that apply modular forms outside of the theory of modular forms. Again, much of the machinery needed for the most interesting (and surprising) applications is there, but not taken advantage of. As we mentioned, the connection between modular forms and elliptic curves which is so central to number theory nowadays is simply omitted. But other “popular” applications of modular forms and Dirichlet series are not mentioned, e.g., the formulas developed by Jacobi, Siegel and others, to count the number of representations of a natural number n as a sum of k squares.

Apostol is an excellent writer of mathematics and the topics that are covered in this book are covered thoroughly in a concise, precise manner. As in the first volume, the writing is characterized by its easy, readable, fluid style. Each chapter is complemented with a nice set of exercises. Unfortunately, though, *Modular Functions and Dirichlet Series in Number Theory* is best defined by what was left out.

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.*