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

I have the earlier two books by Ash and Gross. I particularly like "Elliptic Tales," and so was anticipating this introduction to modular forms. But this book doesn't quite work.The book is divided into three parts -- 1) finite sums, 2) infinite sums, and 3) modular forms. The recurring theme that binds together the three parts is expressing numbers as sums of squares and of calculating the number of ways this can be done. In this connection, Bernoulli numbers (and functions) allow the authors to segue from the first part to the second. The discussion of Bernoulli numbers, the Riemann zeta-function, and generating functions is woefully brief -- considering the conceptual leap required from the first part to the second and considering that the target readership is assumed to have no background in complex analysis. But this is a minor quibble.The third part attempts to introduce modular forms and in this connection discusses SL2(Z), fundamental domains, q-expansions, dimensions of vector spaces of modular forms, Hecke operators, and L-functions. The chapter on applications includes some desultory discussion of partitions that doesn't lead anywhere and a worked example for the number of ways 6 can be expressed as a sum of squares -- yet which doesn't really employ the topics mentioned above. It seems to be a mix of weak heuristic explanation coupled with one solitary example. If the target reader is a scientific layman, I doubt he'll be able to follow the discussion. If a math undergrad, the discussion won't be ample enough, nor will it be precise and structured enough.To be fair, the math is difficult. My contention is that anyone trying to make this palatable to a lay readership will fail. The books I would propose for coverage of modular forms would be:1) Modular Functions and Dirichlet Series in Number Theory, by Apostol,2) Elliptic Curves, Modular Forms, and their L-Functions, by Lozano-Robledo,3) Introduction to the Mathematics of Fermat-Wiles, by Hellegouarch,4) Introduction to Elliptic Curves and Modular Forms, by Koblitz, and5) A First Course in Modular Forms, by Diamond and Shurman.

I liked this book more than other similar books I've tried. Reading it is like having someone explain some math to you personally. No matter how much (or little) you know, there's probably something new in here for you to think about, unless perhaps if you're mentioned in this book's references or are a number theory researcher who would be entirely comfortable talking a piece of chalk in hand after some random person asked "Say, I'd like to know what a modular form is." In particular if you've seen the recent Ramanujan movie you might like this as a slow-starting but then ever-accelerating walkthrough of some of the number theory in it, maybe. By the intermediate value theorem, there's provably a moment at which you have to be learning something.

Avner Ash and Robert Gross have written three roughly a series books ‘Elliptic Tales’, ‘Fearless Symmetry:Exposing the Hidden Patterns of Numbers’ and in 2016 ‘Summing It Up: From One Plus One to Modern Number Theory’.‘Summing it Up’ is somewhat different and less technical than the prior books but gets down to some of the basics of the other two books and is more accessible to a broader audience. It may seem more elementary in some ways but in fact gets behind some of the calculations in Number Theory. The writing and ‘tricks’ remind me of books by Paul Nahin and especially ‘Interesting Integrals’ for showing how to calculate things in different ways and how to get around problems when the calculations you try become a mess or seeming dead-end.

This is an excellent book. It is really worth reading it. It contains very good material on Number Theory starting from basic notions till modular forms and elliptic curves. Some mathematical background is necessary to follow the proofs, which are presented mostly in full details and in a very transparent way. Every mathematician interested in this subject should read it. It is a shortcut to understand the difficulties of some classic problems in number theory. Summarizing, this is a serious maths book written by professionals in the area in a very friendly way, and, most important, there is no bs here.

A very friendly and original introduction to number theory......

This is the third book by Ash and Gross the first two being "Fearless Symmetry" and "Elliptic Tales". Since I have both of those I had to have this. And it's brilliant. All three books are concerned with mathematics and number theory and while their first two books skirted around modular forms this one actually has modular forms as its destination. It is an easy read - you can skip some of the maths if what you want is a general background on number theory, or, you can role up your sleeves and grab a notepad and pen and follow along. Highly recommended.

The authors have produced an amazingly clear introduction to modular forms using very little maths beyond calculus. If you are interested in number theory get hold of this book - you won’t regret it and it may inspire you to tackle more advanced books on the same topic.

A lot of new material that I haven't seen in previous recreational maths books. Plenty of proper mathematics to get your teeth into.

well written and methods easy to follow.

A good book.