How wonderful it is to be able to sneak historical gems into our courses, be they hooks for gifted kids, or orchestration for other courses, typically the meatier ones. Number theory provides a wealth of opportunities for this sort of propaganda for the mathematical cause, given that it is (in)famous for its ability to couch very deep mathematics is seemingly simple terms, at least at first glance. Consider, as a case in point, that ever so innocent question, are there infinitely many primes? Is there any proof more elegant than Euclid’s proof of the infinitude of the primes? Arguably not. Unless maybe it is Euler’s proof of the same fact. Both proofs are incomparably beautiful but very different: they illustrate the fact that in our art, it’s not always that something is proven, but often how it’s proven. The different proofs evince different deep connections with other mathematics --- in other words. what did Euler see? With the benefit of 20-20 hindsight (I guess we can say 2020 hindsight, at least for four more months), what is noteworthy here is the simply stunning quality of Euler’s genius. Ultimately, his proof unearths the number theoretic bombshell that is the behavior of the zeta function at the pole at 1, i.e. the adult version of the fact that the harmonic series diverges.

Well, the cat’s out of the bag now, isn’t it? Euler inaugurated the search for the nature of the distribution of the primes. Soon after Euler, the challenge was taken up by none other than Gauss, who took the question in what appears to be a different direction. Gauss asked what the asymptotic behavior of the distribution of primes would be, and suggested that it should look like \( x/(A \log(x) + B) \) for some \( A, B \): evidence of both Gauss’ genius and his rather different way of looking at things than Euler. Obviously, either way, this is all very deep mathematics: what is really going on here?

Enter Riemann. In what is certainly a contender for the most magnificent paper ever written in mathematics, and also a disturbingly short one (nine pages!), he addressed the question of the distribution of the primes directly in terms of what is now called the Riemann zeta function (pace Euler). In this 1859 paper, “On the number of primes under a given magnitude,” Riemann gave two proofs of its functional equation (the infinite series having been meromorphically continued to the entire complex plane), and then stated and sketched a proof of what is now called the prime number theorem, i.e. the assertion that \( \pi(x) \sim x/\log(x) \). Here \( \pi(x) \) is the player mentioned in the paper’s title, i.e. the number of primes below the real value \( x > 0 \), and we’re dealing with the natural logarithm.

The prime number theorem was proven about forty years after Riemann’s paper appeared, by Hadamard and De la Valée-Poussin, and some decades later it was proven again by Selberg and Erdös whose methods, in contrast to Riemann’s, Hadamard’s and De la Valée-Poussin’s, were “elementary,” which means, primarily, that complex analysis doesn’t get to play.

But, to get back to Riemann, the lead role in this magnificent work of 1859, is unquestionably played by the Riemann zeta function whose series representation, valid for \( \mbox{Re}(s) > 0 \), is \( \zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^{s}} \). Yes, \(s= \sigma + it \) is a complex variable, and this series converges absolutely in the indicated half-plane. The Riemann Hypothesis states that, for the meromorphic continuation of \( \zeta(s) \), which Riemann gives in terms of a complex integral, the so-called non-trivial zeroes all have \( \sigma=\frac{1}{2} \). Apropos, one shows that there are also “trivial” zeroes for this meromorphic function located at the negative even integers, but they’re small fry. And then there is the pole at 1, of course: aye, there’s the rub. Anyway, this question, i.e. the challenge to prove this assertion about the non-trivial zeroes of \( \zeta(s) \), is the hottest problem in our game. It is known that an infinite number of such zeroes lie on this “critical line” but it has not been shown yet that there are no roots (in what is called the critical strip: \( 0 < \sigma < 1 \) off the line \( \sigma=\frac{1}{2} \).

In the book under review, \( \zeta(s) \) appears in the second part (of six), which features, among other things, harmonic analysis; the prime number theorem follows a couple of sections, or chapters, later. I must say that, for me, this second part is my favorite, including, as it does, a great deal of material on those famous generalizations of \( \zeta(s) \), the Dirichlet \( L \)-functions (or \( L \)-series): Dirichlet used them to prove that in every primitive arithmetic progression, i.e. in every set of the form \( \left\{ an+b | n \in \mathbb{Z}, \mbox{gcd} (a,b) = 1 \right\} \), there are an infinite number of primes.

Koukoulopoulos goes on to address such mainstays of number theory as sieve methods (Selberg on p. 213), and, in his twenty-eighth chapter, the very sexy stuff concerning gaps in the sequence of primes. For dramatic effect, we have the still open twin primes conjecture, to the effect that there are infinitely many primes two apart. In this direction, there’s the stunning 2013 work by Zhang that yields that there is some even number \( r \) such that there are infinitely many primes of the form \( p \), \( p + r \), and so the goal is to get \( r \) down to 2. On the other hand, it is trivial to show that for any natural number \( R \) there are prime deserts of length \( R \), just by noting that the sequence of \( R \) successive integers

\( (R+1)! + 2, (R+1)!+3, \ldots, (R+1)! + R + 1 \)

contains \( R \) successive composites. What a wild set the set of primes is! Koukoulopoulos is quite up to date on this exciting material: see p. 300 of the book, where Zhang’s Theorem is introduced (in a far more exciting form than I’ve stated it above: the indicated *lim inf* of the sequence of differences of primes \( r \) apart is shown to be bounded), and variations by Tao and by Maynard are cited, too. A caveat, this kind of number theory is suffused with analysis, primarily the business of special functions, integral transforms, error estimates, growth rate analyses, and so on, which is not everyone’s cup of tea. But for those of us who like this brew, it’s simply marvelous.

It’s clear that Koukoulopoulos had a marvelous time putting together this beautiful material, and producing a very readable and pedagogically sound text (replete with good exercises). The book is well-paced and reads very well. The careful reader, with pencil and paper in hand, keen to do exercises galore and have fun doing so, will learn a lot of beautiful number theory and find out marvels about the secret life of the set of primes: they are elusive but not unyielding.

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