Review of Number Theory: An Introduction via the Density of Primes, second edition

Abstract

There are probably at least as many different approaches to number theory as there are books written about it. Some broad distinctions include those taking an historical versus (say) a purely modern approach, with many gradations in between, or those that are algebraically oriented (e.g., with an emphasis on reciprocity laws, or questions that relate to algebraic geometry), or still others that are more analytic. The book under review is definitely in the latter category. The "message" of the book is in the title, as primes and their density are the principle concern. In accordance with that theme, a highlight of the book is a complete proof of the prime number theorem. However, the theme and its variations are taken as springboards to other important fields, including aspects of algebraic number theory, as well as applications, such as primality testing and cryptography.