This is a wonderful book. While it is dense, sweeping in its scope, and ambitious, it succeeds in what the authors wish for this book, namely, to introduce the committed reader to the modern analytic theory of numbers by means of probabilistic methods. In the span of a mere 115 pages a fearsome panorama of topics is attacked: sieve methods, Riemann's zeta function (de rigeur of course), the distribution of the primes, and then the prime number theorem itself, done a la Daboussi (as opposed to straight Erdos-Selberg). And at the end there is a tantalizing section titled "Major Conjectures." It is accordingly no surprise that a lot is required of the reader: be prepared to work (also see the third caveat, below). However, the beautiful material and the elegant approach and presentation by the authors make it all worthwhile.
A trio of caveats: It is not advisable — in the opinion of this reviewer — to use this book as a pure introduction to analytic number theory. It is far too terse for that purpose. It is more valuable as a (somewhat idiosyncratic) commentary on the central themes of the subject, by authors who are true prostelytes for their cause. Which leads to the next caveat: Tenenbaum and Mendes-France operate from a particular philosophical position, with probability figuring greatly throughout the book. While the book is self-contained it were advisable for the reader to have some training in that area. And the third caveat: "We have... chosen to follow a different path with the deliberate intention to aim a little higher than is usual in a work designed for a broader readership... We have sometimes preferred a short calculation... to a long explanation, and the style is purposely condensed, even to the extent of being in some paces elusive..." True. The reader should take the authors' advice and read synthetically, not analytically, being willing to go back later to dot i's and cross t's. But this is good for the soul.
Finally, the authors take some pains to include topics that are rather sexy these days, e.g. cryptography and some computer graphics vis a vis the stochastic distribution of the primes. But even aside from these, this is a thoroughly modern book, in the best sense of the phrase: it brings a beautiful collection of results in analytic number theory together around the unifying theme of probabilistic methods, touching on some marvellous avant garde stuff. It would be a fine text to use for a senior or even a graduate seminar in analytic number theory.
Michael Berg (firstname.lastname@example.org) is professor of mathematics at Loyola Marymount University in Los Angeles, CA. His research interests are algebraic number theory and non-archimedian Fourier analysis.