Number Theory

George Andrews, of Penn State, a mainstay of the number theory scene for many years now, is happily still going very strong at age 76. Here, for instance is his itinerary of talks given just in this calendar year: http://www.personal.psu.edu/gea1/pdf/2015Talks.pdf. This admittedly small sample does bear out the proposition that Andrews’ interests and expertise include combinatorics, additive number theory, and the theory of partitions, and this point is amplified — in spades — by his list of publications (cf. his web page). Indeed Andrews has long been one of the major players in connection with all things Ramanujan, in particular the Rogers-Ramanujan identities and the various activities surrounding Ramanujan’s lost notebooks, and he has contributed in major and varied ways to the theory of partitions, to name one obvious example of his prolific activity.

The book under review, an introductory level text on number theory at the undergraduate level, accordingly benefits from what only number theory can offer at this level, and what only an expert like Andrews, with his orientation in the direction of combinatorics, can bring about. Published in 1971, it has been around for quite a while now, and it is a wonderful benefit that Dover is now launching this book at the customary unbeatably low price; well, to be precise, Dover first launched the book in 1994, but it is still (?) only $14.95. It is a superb text for a rookie course in number theory, or even an introduction to pure mathematics aimed at propelling tyros into the thick of it all with zest and style.

The sequence of topics Andrews hits in his book is partitioned into five parts: “multiplicativity — divisibility,” featuring “FT Arithmetic” and the theory of congruences and capped off by good stuff about \(\pi(x)\); “quadratic congruences,” focused on Gauss-Euler quadratic reciprocity (i.e. the inner life of the Legendre symbol); “additivity,” where we encounter the theory of partitions and the vaunted Rogers-Ramanujan identities; and finally “geometric number theory” which focuses on Gauss’s circle problem and Dirichlet’s divisor problem.

The book comes equipped with a number of excellent and useful appendices, but even more importantly, Andrews liberally peppers his text with exercises aimed at exposing his readers and students to arithmetical calculations of varying (i.e. locally gently increasing) degrees of difficulty, as well as equally well orchestrated propositions to prove. Indeed, these fabulous exercise sets, offering things for students of different abilities, together with the fact that Andrews’ prose is crystal-clear and to the point, make the book ideal for use in the kind of course it is intended for: a true introduction to what Davenport called “the Higher Arithmetic.”

I haven’t had occasion to teach number theory at this level for a number of years, but I look forward to doing so again. When I do, I’ll use this book as my text.

Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.