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Famous Functions in Number Theory

Bowen Kerins, Darryl Yong, Al Cuoco, and Glenn Stevens
American Mathematical Society and IAS/Park City Mathematics Institute.
Publication Date: 
Number of Pages: 
IAS/PCMI-The Teacher Program Series
Problem Book
BLL Rating: 

The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Hema Gopalakrishnan
, on

This problem-based book on Number Theory focusses on discovering fascinating properties of multiplicative functions. This is one of three books based on a three-week intensive summer course in the Summer School Teacher Program (SSTP) at the Park City Mathematics Institute. The book is divided into several sections: Fifteen Problem Sets, Facilitator Notes, Teaching Notes, Mathematical Overview, and Solutions.

The key concepts explored in the problem sets are multiplicative functions, modular arithmetic, and infinite series and products. Each of the fifteen problem sets has an Opening problem. The remaining problems are classified as “Important Stuff”, “Neat Stuff” and “Tough Stuff”. Most of the key concepts are introduced in the first few problem sets in the Important or Neat Stuff. Problems that are important but require more time for understanding are revisited in multiple problem sets.

A variety of multiplicative functions are introduced throughout the course. In particular, each of the first five problem sets begin with a deep exploration of a different multiplicative function including the famous sigma function and Euler’s phi function. The Opener as well as the first few problems in these sets guide readers to work through many numeric examples for each of these functions, organize information in tables, identify patterns, devise formulas that help to compute the value of the function quickly for many natural numbers, and find a recurring similarity in the behavior of these functions.

Modular arithmetic is introduced via numerical examples as well as in the disguise of multiplicative functions in Problem set 3. It would be interesting to see this connection made explicit. Solving equations modulo composite numbers is an important goal; the use of charts to organize the numbers modulo n and look for solutions modulo relatively prime factors of n is very helpful to provide insight into the method. Also introduced in this set is the concept of a sum of an infinite series. A series of algebraic and geometric explorations in this and subsequent problem sets are used to help conclude that the harmonic series diverges, whereas the sum of the reciprocals of squares of natural numbers is finite. This leads further to a study of infinite products of infinite series in later problem sets.

One of the most interesting concepts in this book, introduced in Chapter 4 and setting the tone for the remaining problem sets, is that of a “child” function and “parent” function of a function. Although this is not a new concept, using the vocabulary of “child” and “parent” makes the problems relatable and motivating. The process of finding these functions gives rise to other famous functions, such as the tau function and the Möbius inversion function, and leads to important connections between multiplicative functions as one can be the child, grandchild, parent, or ancestor of another. Moreover, using basic algebra to multiply infinite series term by term helps to further understand the parent child relationship between multiplicative functions. It is quite surprising and beautiful that expanding powers of certain polynomials helps to easily gather data on the number of ways a natural number can be expressed as a sum of k squares. These concepts pave the way to a conjectural rule for the number of ways in which a natural number can be expressed as a sum of two squares. This is one of the capstone results in the course.

The Facilitator’s Notes section will be valuable for those teaching with this book for the first time. It gives a brief description of the goals of each problem set along with a brief explanation of each problem in the “Important Stuff”. This includes the authors’ observations on the ease or difficulty experienced by participants in their course for some problem sets, the problems that must be explored within a set and those that can be skipped or worked on when they recur in another problem set, suggestions for when a whole group discussion would be appropriate and specific insights into solutions. There are a few discrepancies, as some of the problem numbers and the terminology used in the book differ slightly from the course website: For instance, the notes on Problem set 4 state that Problem 11 in this set depends on Question 5 of Problem set 3, which is not true in the book but true on the website. Also there is reference to “boxed problems” in the Facilitator’s Notes. These are typically the Opening problems in the book. But these are minor errors and can be clarified by referring to the website.

A calculator for symbolic computations is appropriately recommended for a few problems in the book, to avoid straightforward but tiresome calculations by hand. The section on Teaching Notes discusses the main ideas in each section that could be summarized as a whole group. The website stated above has other resources that maybe useful to the instructor such as overlay transparencies and calculator commands. The section on Mathematical Overview will be very helpful to anyone who is curious to know more about the theorems that motivated the problems and concepts in the book, especially that of a “child” and “parent” function. This book is written mainly for instructors, as the sections on Facilitator Notes and Teaching Notes are not relevant to participants of the course.

As intended the problem sets serve to deepen the mathematical content knowledge of precollege teachers. But they will also appeal to a wider audience of students and faculty. High school teachers and undergraduate students and faculty who are already familiar with second semester calculus and modular arithmetic will enjoy solving the “Important” problems as well as some of the “neat” or “tough” problems in each set. The problem sets can serve as a mathematics content course for students in a graduate program in mathematics education. Middle school math teachers can also successfully learn from this book: the concepts in the “Opener” and “Important Stuff”, which form the core of each problem set, are developed solely through numerical examples, patterns, and critical reasoning, without emphasizing formal proof. Therefore, this book can serve as a resource for instructors of Math Circles and Math Teachers’ Circles. To a motivated group of participants eager to learn simple concepts about numbers deeply through inquiry and perseverance, this book will be rewarding. It is filled with amazing connections!

Hema Gopalakrishnan is associate professor of mathematics at Sacred Heart University and leads the Fairfield County Math Teachers’ Circle in Connecticut.