NB. I was sent this book as a review copy.

From Princeton University Press

I have to admit that I felt very skeptical when I started reading this book. In the prologue it is stated that the book is aimed at enthusiastic readers of mathematics with an AP level of high school maths. Then, diving into the book one sees what looks at first sight like a pure maths textbook at graduate level. But Paul Nahin isn’t one to pull a fast one like that, so I read further. In fact, I raced through it, hugely enjoyed it, and in the end agree with Nahin that someone with a US AP level of high school maths, or here in South Africa a confident first year undergraduate could actually understand everything in the book.

The book is not written as a textbook on mathematics, much as it might look like one, but rather it is taking an historical path through the investigations into the mysteries of zeta(3). The subtitle of the book is a provocative one, but he takes pains to point out that by this statement he means problems which are easy to understand, but very very hard to solve, and have some connection to the real world, which indeed the zeta functions do have within physics and computer science (in particular cryptography) and arguably other areas as well. He does mention the four-colour problem and the Collatz conjecture, though it’s unclear what the utility of the Collatz conjecture is in the real world (though it is wonderfully simple and mysterious too).

The premise of the problem is very simple. Given:

\zeta(k)=1+\frac{1}{2^k}+\frac{1}{3^k}+\frac{1}{4^k}+...

and that we can write very simple expressions for this when k is a positive even integer, can we do so for k=3? Well, the answer so far of course is no, we can’t, but there is still a great deal to be learned by exploring different ways to represent the zeta functions, through integral representations, Fourier series, and via the connection with Gamma, Beta, and Riemann zeta functions.

All of these topics are then given a particular purpose in trying to understand more about this mysterious quantity, and while there is of course more to say about each of them than can be in such a short book, the book does manage to cover an incredible amount of ground on a wide range of topics.

I would highly recommend this book as a present to any student who seems to have an interest in diving further into mathematics than they may be getting in their first year university courses and won’t be scared off by the quantity of equations. I thoroughly enjoyed this book!

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