A Course in Arithmetic (Graduate Texts in Mathematics, Vol. 7) (Graduate Texts in Mathematics, 7)

The book is divided into two parts -- algebraic and analytic. I've only worked through the analytic part. Anything by Serre is worth its weight in gold and this book is no exception; everything Serre covers is of the utmost importance. But Serre's style is extremely condensed and spare, and he makes no concessions to the reader in terms of motivation or examples. I can't digest more than half a page of Serre a day; however if one wants to understand the structure of a theory, Serre is ideal.I worked through "A Course in Arithmetic" over a decade back. As I recall I covered Riemann's zeta function and the Prime Number Theorem, the proof of Dirichlet's theorem on primes in arithmetical progressions using group characters in the context of arithmetical functions, and some of the basic theory of modular functions. All of this material is also covered in Apostol's two books on analytic number theory ("Introduction to Analytic Number Theory", and "Dirichlet Series and Modular Functions in Number Theory"); Apostol goes further than Serre in the analytic part -- which is only to be expected since he is devoting two whole texts to the subject.

Serre is a great writer. He can be terse, and thus reading him is demanding, but he is careful and is not terse out of laziness, so reading him is possible. You won't invest time reading a chapter and then find mistakes or omissions or get hopelessly confused and then stop reading the book in frustration. I think this book would be excellent for an undergraduate reading course; probably the book is too long for one semester, but it is nicely divided into two parts, algebraic and analytic, that can be read separately. I like the chapter on p-adic fields; to digest it one needs to be familiar with the ideas of projective systems and exact sequences. The chapter on modular forms is a self-contained introduction to the topic. However, as an introduction to modular forms I prefer Bump's introduction, in the first chapter of his "Automorphic Forms and Representations", and when I first learned about modular forms I used both Serre and Bump.

I recently bought a copy on Amazon of this (terse but excellent) classic book. On the copyright page it says "Printed and bound by R. R. Donnelley and Sons, Harrisonburg, VA." The print quality is noticeably poor -- the text is gray, and the detail is coarse, like an old low-resolution laser printer. There are also a few little blips where, for example, part of a letter will fail to print. At roughly $1 per two-sided page, you would expect better. I don't know if all copies look like this, in all other respects it appears to be a normal GTM. I wonder if Springer might be using some kind of print-on-demand service for older, lower volume titles that might otherwise fall out of print.

J.P. Serre, the author, is one of the greatest mathematicians of the past half century.He is also a renowned expositor of advanced mathematics. The book inquestion is a valuable reference in a very active area of modern mathematics.PK

A classic. Also fun to keep around to hear people say "I thought you had a PhD. Shouldn't you know arithmetic?"

This book is very elegant, a pleasure to read, but not a great textbook -- after reading you are likely not to remember anything other than having enjoyed it (this is particularly true of the proof of Dirichlet's theorem). For actually learning to work in the subject (of analytic number theory), Davenport's book  Multiplicative Number Theory  is VASTLY superior.

Here I am trying to help out some of the more clever upcoming 3rd graders. Their parents contacted me asking about some math books their kids could do over the summer to keep them sharp and ready for next year.I said get a book on Arithmetic, and looked up this one on Amazon and sent them a link to buy it.Well long story short, the parents are coming unglued, griping about common core and how confusing all this stuff is and how they never had to do this when they were kids [what a bunch of "math is hard" whiners]. They claim they have no idea what the book is about, and their little genius snowflakes are crying because they can't go to Disneyland until they finish this book. I mean for Pete's sake they should have had this stuff memorized by the end of 2nd grade.Need to go talk to my 2nd grade colleague and found what she was doing all day (I thought I smelled liquor on her breath) since these supposed "smart kids" are dumber than a box of rocks if they can't handle arithmetic...geesh parents these days.

wow.

Of course you buy a book like this for the content but the quality of the printing suffers a lot when compared to other Springer publications. I'm not sure when this started but the pages are starting to look like photocopies. A little disappointing.