Number Theory Revealed: An Introduction, by our author Andrew Granville, was expanded into Number Theory Revealed: A Masterclass, which has now been followed by the book under review, Analytic Number Theory Revealed: A First Guide to Prime Numbers.  The Masterclass is suggested as providing “the necessary background” for the present book.  Looking at the contents of the Masterclass, one is struck by the wealth of material covered, ranging from beautiful to gorgeous.  Here are six samples (6 being, after all, perfect): solving the general cubic, followed by solving the general quartic; the dynamics of the Euclidean Algorithm; the group of rational points on an elliptic curve; Pythagorean triangles of area 6; Waring’s Problem.  One notes the span of beautiful topics covered, from what has traditionally been called elementary number theory, to elliptic curves, for example.  For someone of my vintage, the book is reminiscent of Hardy and Wright, An Introduction to the Theory of Numbers, dating to 1938.  Of course, Granville has a lot more number theory to choose from, e.g. the business of elliptic curves and their rational points, with Mordell’s Theorem given its proper due, but there’s a lot more, of course.

Does one need that much preparation to tackle the book under review?  No, but the Masterclass is clearly a smorgasbord of gourmet dishes rich enough in variety to please every gourmand.  It’s sufficient but not necessary.  To wit, Granville’s Analytic Number Theory Revealed is devoted to sieve methods for the primes, the evergreen Prime Number Theorem, Dirichlet’s Theorem on primes in arithmetic progressions, and as the book’s last chapter, “A dozen and one different directions,” a dozen and seven topics of independent interest.  Here are, again, six: Selberg’s small sieve, primes in short intervals, primes missing digits, analytic continuation for certain Dirichlet series, different proofs of the PNT (yes, the Prime Number Theorem), and to cap it all off, a primer on the circle method (of Hardy and Littlewood: the more things change, the more they stay the same…).  But there’s a question that’s been begged, namely, what does one really need to take a deep dive into this true-blue analytic number theory?  Granville’s chapter, “Background in analysis,” points toward the answer: Fourier series and analysis, complex analysis, and then the crowning number theoretic example of analytic continuation, that of the Riemann zeta function.  So, is this enough?  Only working modulo mathematical maturity (admittedly not a well-defined notion, but to be sure, one knows it when one sees it) and a willingness to do so.  But once these conditions are met, the experience of studying this book is certainly worth it.

Granville’s approach is compact, clear, and thorough.  There is a great deal of history woven throughout the text, as per, for example, two appearances of the duo of Hardy and Ramanujan on pp. 64-65, in connection with asymptotic considerations for the Prime Number Theorem.  Or as far as pedagogical themes go, again kudos to Granville: see, e.g. Section 26.1: “Dirichlet’s plan,” inaugurating (what else) “Primes in arithmetic progressions.”  À propos, in an earlier age, a decade or more before my elevation to emeritus status and my being put out to academic pasture, I had occasion to supervise a senior student in a thesis on Dirichlet’s Theorem, and looking back, I wish Granville’s book had been around then.  I used other fine sources, to be sure, including Serre and Apostol, but the unity and pitch Granville presents would have suited her better, I think.

Anyhow, I am obviously thoroughly impressed with this book.  I intend to take some deep dives myself, just because it’s all so beautiful and so well done.  I can’t recommend it highly enough.

Oh, yes, lest I forget, Granville also has a bunch of terrific (also compact) biographical sketches of some of the cast.  Here are (what else?) six: Gauss (non-negotiable), Ramanujan, Selberg, Jacobi, Riemann (my favorite mathematician of all time), Dirichlet.  There are more, but I’ll leave them for you as a surprise for when you read this wonderful book.  Officially, Granville intends the book for advanced undergraduates and rookie graduate students.  Fair enough.  But it’s so good!  Even an old fogey like me, having done at least a chunk of this material for over fifty years, can look forward to a lot of fun taking the aforementioned deep dives.

Post scriptum, do not miss page v and the revealing photograph: something along the lines of Granville revealed.

Michael Berg is an emeritus professor at Loyola Marymount University, retired after three decades of teaching mathematics, some Thomist philosophy, and judo (eventually from his home dojo). He continues his long-time research in the area of analytic methods in number theory, focusing in particular on higher reciprocity laws for number fields.