# Summing It Up: From One Plus One to Modern Number Theory

This book is the third in a trilogy by these authors who want to introduce number theory to the `mathematically literate reader'. The first two books were *Fearless Symmetry: Exposing the Hidden Patterns of Numbers* (2006) and *Elliptic Curves: Curves, Counting and Number Theory* (2012). The first dealt with Diophantine equations and Fermat's Last Theorem (FLT), and the second with elliptic curves and the Birch-Swinnerton-Dyer (BSD) conjecture. Both of these somehow ran ashore at the end when bumping into modular forms which were not fully explained. This third book *Summing It Up: From One Plus One to Modern Number Theory* has as its main objective to introduce the reader to these modular forms and their applications. Like the previous books it has three parts. Parts one (finite sums) and two (infinite sums) are further motivations and at the same time also preparations for the main part three about modular forms

Number theory in its simplest form is easy to understand even for the man in the street and innocent looking questions or conjectures can be formulated, and yet answering or proving them is extremely difficult and may take even centuries and the brightest mathematicians to answer them. The good thing about this is that, in trying to answer these questions, whole new mathematical areas are explored that lead us far away from the simple question about integers. Modular forms is such an instrument from complex analysis with major application in number theory, but that also appear for example in string theory and algebraic topology.

It is not a simple task to explain to the lay person that a modular form of weight *k* is a holomorphic function of the upper half plane of the form $f(z)=a_0+a_1q+a_2q^2+\cdots$, $q=\exp(2\pi i z)$, bounded as $\mathrm{Im}(z)\to\infty$ and that satisfies $f(\gamma(z))=(cz+d)^kf(z)$ for any $\gamma$ in the modular group, i.e., $$\gamma\in\mathrm{SL}_2(\mathbb{Z})=\left\{\left[\begin{array}{cc}a &b\\c&d\end{array}\right]:a,b,c,d\in\mathbb{Z}, ad-bc=1\right\}\quad\text{defining}\quad\gamma(z)=\frac{az+b}{cz+d}.$$ And yet, the authors succeed in gradually leading the reader to this level and connect this to elliptic curves and number theory. They have to guide the reader through complex analysis, hyperbolic geometry, introduce the concept of groups, fundamental domains, and analysis of the modular group and their traditional generators and congruence subgroups. To make sure the reader is following up to that point, it requires strong motivation, and of course the reader should not be completely unfamiliar with mathematics. In fact the mathematical skills required are building up as one is reading on. With only a first calculus course in your backpack, it can be done, but you will need some patience and perseverance to reach the end.

Part 1, requires only high school algebra and some geometry. This part on finite sums introduces modulo calculus and the type I (equal 1 modulo 4) and type III (equal 3 modulo 4) numbers, which are important to know when a positive integer is the sum of two squares. The latter problem is then gradually generalized to: find out how many ways there are to write a positive integer $n$ as a sum of $k$th powers. Another topic treats summation techniques and formulas to add a finite number of $k$th powers, including binomials and Bernoulli polynomials.

For the second part the reader needs to know differentiation, infinite series, and Taylor expansion as it can be found in a first year calculus course. Besides the infinite sums (and products), the reader is also is prepared for part three with complex numbers and some complex functions, especially the symbol $q=e^{2\pi i z}$ is important in for modular forms. Furthermore it also introduces analytic continuation, the zeta function, generating functions and Dirichlet series.

The third part is the most important and takes about half of the book. Since the previous part introduced the necessary material, hence provided it sank in well enough, no extra knowledge is required. As mentioned above it leads the reader to the concept of a modular form. But it goes beyond just the definition. What if subgroups of $\mathrm{SL}_2(\mathbb{Z})$ are considered, and how do these relate to the weight *k* of the modular form? And how are modular forms related to generating functions for a sequence of numbers? For example the sequence $\{p(n)\}_{n=1,2,...}$ where $p(n)$ represents the number of partitions of $n$, i.e., the number of different ways in which $n$ can be written as a sum of positive integers. The generating function for this sequence can actually be written as an infinite product, which, like in the case of the zeta function, makes it interesting. Here the link with previous topics can be made: consider a sequence $\{a(n)\}$ as considered in part 2. I.e., $a(n)$ expresses the number of different solutions to a problem depending on a positive integer $n$. If these are the coefficients in a generating function or Dirichlet series, then modular forms can be used to investigate their properties. But not only the applications in number theory, also applications in other areas (some of them are mentioned in the trailing chapter) make modular forms an interesting research area *an sich*.

There is some mathematics indeed: there are definitions, theorems, proofs, but the most complicated proofs are left out. There are even some occasional exercises for the reader to work out, but the text is not really structured in a strict textbook kind of way. It is more a written-out oral presentation with a lot of explanation and intermittent questions like: "such and such is true. Why? Well remember that..." or "What happens now if..." or "Is that good enough? Yes!" etc. There are also reasonings that simulate the reader's thoughts in the style of "Let's try ... But that does not work if... So we need to..." until all pitfalls are removed. Only after this `trial and error' intro the eventual correct definition is formulated. Nevertheless, it is not the hand-waving kind of popularization. There is true rigorous mathematics and the reader should be prepared to swallow it. Anyway the authors did a remarkable job in making some aspects of modern number theory very accessible to readers with only a minimal knowledge of mathematics, say a student who had a first calculus course. However, also mathematicians who do not have number theory as their main focus will enjoy this book although they will probably skip some of the sections dealing with the introduction of elementary topics such as summation formulas, groups, complex numbers, or vector spaces. They will learn how complex analysis and group theory (joining forces in modular forms) are nowadays working tools to tackle some questions in modern number theory.

**Submitted by Adhemar Bultheel |

**5 / Jul / 2016