Customer Reviews

A Primer of Analytic Number Theory: From Pythagoras to Riemann

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A great book for senior undergraduates in mathematics or anyone with some background in calculus and complex numbers. Proofs are at a level where a careful reading makes them clear, and the author tells the reader when he is not being rigorous. Historical background and logical development of topics makes this a good read too. Most surprising to me was how the author tied in topics from prior chapters into later chapters--he didn't just jump from one topic to the next willy-nilly, but made the book flow as a whole. Problems given to the reader were helpful though sometimes too hard for me, a math major.

There usually seems to be a pretty big gap between the math background needed to understand books on elementary number theory and what's needed to understand most books dealing with analytic number theory, and this book does a good job of making that gap seem smaller. The writing feels a bit like Silverman's "Friendly Introduction to Number Theory" and Derbyshire's "Prime Obsession." There are plenty of experiments for Mathematica and Maple. I could see this book being used in an undergraduate number theory class. The book doesn't assume any familiarity with complex variables. If you can integrate and aren't too afraid of series or logarithms, this book should be no problem.

The book goes over multiplicative functions, Mobius inversion, the Prime Number Theorem, Bernoulli numbers, the Riemann zeta function (and its value at 2n, its analytic continuation, its functional equation, and the Riemann Hypothesis), the Gamma function, Pell's equation, quadratic reciprocity, Dirichlet L-functions, elliptic curves (including their L-functions and the Birch and Swinnerton-Dyer conjecture), binary quadratic forms, and an analytic class number formula for imaginary quadratic fields.

I recommend this book to anyone who can read; and for those who can't read, this book is good motivation to become literate.

A little background on me. I have just finished my freshman year of high school, and this was my first book on number theory. However, I have read many other math texts. In the beginning of the book there are some new concepts introduced, but they are not too hard to understand. The middle is refreshing as it involves a lot of calculus, which the student is most likely familiar with. The latter part consists of a variety of new ideas, and the theorems can get quite lengthy. I do not fully understand all of them myself. The book is well written and also includes the history of many mathematical problems.

This is definitely one of the delightful math books to read. The material is so well organized that it flows very nicely. This book provides a very gentle introduction to such topics as Zeta function and Prime Number Theory.