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Prime Numbers and the Riemann Hypothesis

Barry Mazur and William Stein
Cambridge University Press
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The Basic Library List Committee strongly recommends this book for acquisition by undergraduate mathematics libraries.

[Reviewed by
Mark Hunacek
, on

This is a delightful little book, not quite like anything else that I am aware of. Its goal (to quote the authors) is to “explain, in as direct a manner as possible and with the least mathematical background required, what [the Riemann hypothesis] is all about and why it is so important.” This goal has been achieved, but it should be understood that the phrase “least mathematical background required” is not a synonym for “no mathematical background beyond high school”. The authors have (very wisely) not succumbed to the temptation of assuming that they can make students who have just taken a year of calculus, and who don’t even know what a complex number is, conversant with the Riemann Hypothesis (RH). Instead, the requirements for fully understanding the book rise as the text proceeds, and to really follow the entire contents, a background roughly comparable to an undergraduate senior mathematics major is probably necessary. A course in complex variables, for example, would be very valuable for understanding the latter part of the book.

The Riemann Hypothesis is perhaps the most important of the currently unsolved problems in mathematics; it was one of the problems discussed by Hilbert in his famous 1900 address to the International Congress of Mathematicians, and it is also one of the seven Clay Institute Millennium problems (with a million dollar award for its solution). Yet most undergraduate mathematics majors graduate without knowing anything at all about it, and I suspect that even a number of faculty members probably are not aware of just why this conjecture is significant. Indeed, until quite recently, the only thing I knew about the problem was that it involved the roots of the Riemann zeta function, and I had no idea why anybody would care about where these roots lie. It turns out, however, that RH has deep connections with the distribution of prime numbers, and it is a discussion of these connections that constitutes most of this book.

The statement in the first paragraph above that this text is “not quite like anything else” probably requires some explanation, because there are certainly other books that attempt to make the Riemann Hypothesis comprehensible to a broad audience. Perhaps the best-known example of this genre is Derbyshire’s highly regarded Prime Obsession, but the book now under review is somewhat different than Derbyshire’s book. Every even-numbered chapter of Prime Obsession is historical in nature: Derbyshire discusses the personalities of the people involved in the Riemann Hypothesis, with generous helpings of historical anecdotes (the Hilbert-at-the-gravesite story, for example) added. The odd-numbered, non-historical chapters are mathematical in nature, but the mathematics is deliberately kept at a low level; indeed, Derbyshire originally intended to avoid using calculus at all, and when that proved impossible, he used it as sparingly as possible. Mazur and Stein, by contrast, are more interested in the mathematics than in the personalities involved, and the level of mathematics here is higher: they use calculus throughout a lot of the book, as well as even more sophisticated mathematics, such as Fourier series and distributions.

The book is divided into 38 chapters, all of them bite-sized: some are only a page or two long, very few are more than five. These chapters are, in turn, grouped into four parts. Part I, which amounts to about half the book, introduces the Riemann Hypothesis, expressed not in terms of the Riemann zeta function but instead (in two separate but equivalent ways) in terms of prime numbers. After a discussion of just what prime numbers are, and some of their basic properties, the authors quickly get to the question of trying to determine how regularly they appear in the set of all positive integers. The logarithmic integral function \(\mathrm{Li}(x)\) is defined, as is the function \(\pi(x)\), which counts the number of primes that are less than or equal to the positive number \(x\). One version of the famous Prime Number Theorem (PNT) asserts that these two functions are asymptotically equal: their quotient approaches the limit \(1\) as \(x\) approaches infinity. However, the PNT does not tell us that the difference \(\mathrm{Li}(x) - \pi(x)\) is small, and in fact one formulation of the Riemann Hypothesis is the assertion that \(\mathrm{Li}(x)\) is, to within square-root accuracy, equal to \(\pi(x)\). A second formulation of the RH is then obtained by modifying the function \(\pi(x)\) to a function that (if and only if RH is true) is equal to within square root accuracy to the nice, simple function \(f(x)=x\). All of this is explained slowly, carefully and clearly, with lots of charts, figures and numerical data to help illustrate the ideas. (The book is printed on attractive glossy paper, making it a pleasure to read.) All of this is very well done indeed, and a student who reads just this part of the book will be rewarded with some very interesting mathematics. But the real magic of the RH is yet to come, and that discussion comprises the rest of the book.

Part II consists of half a dozen chapters discussing distributions, Fourier series and Fourier transforms. This material, in turn, is then put to work in Parts III and IV of the book, where, in a way too complicated to describe here, a distribution \(\Phi(t)\) is defined by making a number of modifications to the function \(\pi(x)\). Using this distribution, a discrete sequence of positive real numbers called the Riemann spectrum is defined as well. It turns out that the process can be reversed and that (assuming RH) these numbers can, by a Fourier-like analysis, be used to reconstruct the function \(\pi(x)\); this was the content of Riemann’s ground-breaking 1859 paper. Part IV of the book also introduces the zeta function, the zeros of which are related to the Riemann spectrum: the final formulation of the RH given in the text is that the nontrivial zeros of the zeta function all have real part ½ (and the imaginary parts are nothing more than the numbers appearing in the Riemann spectrum, along with their negatives). Hence, the truth of RH establishes a “nice” distribution of the primes; if RH is false, their distribution is more complicated.

This is, of course, very difficult mathematics, and the exposition does get a bit technical, especially in the last half of the book. But the authors have done a good job of conveying the ideas behind the program. They write in a clear, engaging way, frequently summarizing what has been done and previewing what is to come. The upshot is that even a student who can’t follow every detail will likely emerge with a good intuitive sense of what makes RH so important.

This book has an interesting publication history. Over the course of a decade, the authors devoted one week per year to writing it, posting the results on the internet and inviting comments. As a result of this frequent feedback, the book seems to be quite free of errors, though I did notice a couple of things that merit comment.

On page 32, for example, there appears a picture of Gauss, underneath which the years of his birth and death are listed as 1824 and 1908, respectively; now, I confess I did not know the exact dates of Gauss’ birth and death when I read this, but I did know that Gauss did not live into the 20th century. The correct dates are 1777 and 1855.

Also, as previously noted, a recurring theme throughout the book is the statement of different (but equivalent) formulations the Riemann Hypothesis. Several different ones are set out in colored boxes: the first on page 41, the second on page 54, and a fourth on page 123. People with OCD tendencies (a group that apparently includes me, since I noticed this) will wonder where the third formulation is; I couldn’t find one, certainly not one that was nicely boxed like the three above.

Third, the authors’ definition of the Bernoulli numbers on page 120 is, I think, nonstandard; under the definition I learned (and which is apparently still standard, as that is the definition that appears, for example, in the very recent Summing it Up by Ash and Gross), the Bernoulli numbers \(B_4, B_8, B_{12}, \dots\)are all negative, whereas the authors have defined them so that all the Bernoulli numbers are positive. This, in turn, causes problems with the formula the authors give on the very next page for \(\zeta(2n)\): that formula involves a term \((-1)^{n+1}B_{2n}\) and hence, if all the Bernoulli numbers are positive, will produce a negative value for \(\zeta(2n)\) whenever \(n\) is even, which of course clearly contradicts the definition of the zeta function.

Finally, though not an error or typo, I should note that I thought the Index could be improved; there was no reference to the logarithmic integral function \(\mathrm{Li}(x)\), for example, despite the fact that this function plays a major role in the book. And although some books, including Derbyshire’s, are mentioned in the text itself, a bibliography, especially an annotated one, would also have been nice.

But let’s not end this review with quibbles. This book is a splendid piece of work, informative and valuable. Undergraduate mathematics majors, and the faculty who teach them, should derive considerable benefit from looking at it.

Mark Hunacek ( teaches mathematics at Iowa State University.

The authors have created an errata page for the book.

1. Thoughts about numbers
2. What are prime numbers?
3. 'Named' prime numbers
4. Sieves
5. Questions about primes
6. Further questions about primes
7. How many primes are there?
8. Prime numbers viewed from a distance
9. Pure and applied mathematics
10. A probabilistic 'first' guess
11. What is a 'good approximation'?
12. Square root error and random walks
13. What is Riemann's hypothesis?
14. The mystery moves to the error term
15. Césaro smoothing
16. A view of Li(X) – π(X)
17. The prime number theorem
18. The staircase of primes
19. Tinkering with the staircase of primes
20. Computer music files and prime numbers
21. The word 'spectrum'
22. Spectra and trigonometric sums
23. The spectrum and the staircase of primes
24. To our readers of part I
25. Slopes and graphs that have no slopes
26. Distributions
27. Fourier transforms: second visit
28. Fourier transform of delta
29. Trigonometric series
30. A sneak preview
31. On losing no information
32. Going from the primes to the Riemann spectrum
33. How many θi's are there?
34. Further questions about the Riemann spectrum
35. Going from the Riemann spectrum to the primes
36. Building π(X) knowing the spectrum
37. As Riemann envisioned it
38. Companions to the zeta function.