B OOKREVIEW Prime Numbers and the Riemann Hypothesis Donal O’Shea Communicated by Harriet Pollatsek founder of SageMath, a powerful open-source mathemat- Prime Numbers and the Riemann Hypothesis ical software system, and CoCalc. So, if anyone could Barry Mazur and William Stein write an account of this sort, it is they. Nonetheless, the Cambridge University Press; 1st edition (April 11, 2016) goal they set themselves is audacious and highly non- 150 pages, $20.49 trivial. Even today, Riemann’s paper is tough going for ISBN-13: 978-1-1074-9943-0 those without specialized knowledge in the area. Writing mathematically meaningful, but readily comprehensible, text about a highly technical subject for serious readers Introduction is an undertaking not to be underestimated. When that The centenary of Hilbert’s problems and the announce- subject lies deep enough to have consequences in many ment of the Clay Institute’s millennial prize problems re- other areas, that difficulty is compounded. sulted in a number of very good trade books about the Mazur and Stein succeed brilliantly. The first half of Riemann Hypothesis. In their book Prime Numbers and their book assumes no knowledge of calculus, the next the Riemann Hypothesis, Barry Mazur and William Stein third requires some knowledge of real differential calcu- set themselves [p. vii] a different goal: lus, and the last sixth some complex analysis. I cannot A reader of these books will get a fairly rich think of an interested reader who will not profit from this little book. It will fascinate anyone who enjoys mathemat- picture of the personalities engaged in the pur- ics. It provides amateurs and students with an attractive suit [of the Riemann Hypothesis], and of related and accessible entry to a compelling, but difficult realm mathematical and historical issues. That is not of mathematics. Professional mathematicians and mathe- the mission of [this book]. We aim—instead—to matical scientists will love it—so will electrical engineers explain, in a manner as direct as possible and and computer scientists. It deserves to become a classic. with the least mathematical background required, what this problem is all about and why it is so Riemann and His Hypothesis important. In 1859 at age thirty-two, Bernhard Riemann was ap- Mazur and Stein are distinguished mathematicians who pointed to the chair at the University of Göttingen have made significant contributions in areas the Rie- previously held by his teachers Gauss and Dirichlet. mann Hypothesis touches. Both care deeply about The death of Dirichlet, to whom Riemann was very close, teaching. Mazur writes like an angel, and Stein is the had left the chair vacant. For Riemann, the appointment would have been bittersweet. He desperately needed Donal O’Shea is professor of mathematics and president of New the income and recognition, but Dirichlet was one of College of Florida. His email address is [email protected]. his closest friends, and one of the few who understood For permission to reprint this article, please contact: him and his mathematics. Shortly afterwards the Prus- [email protected]. sian Academy of Sciences in Berlin elected Riemann as DOI: http://dx.doi.org/10.1090/noti1700 a corresponding member, and Kronecker encouraged August 2018 Notices of the AMS 811 BOOK REVIEW him to submit a paper describing his research. Whether the seven “millennial” problems for the solution of each by design or serendipity, Riemann had been exploring of which the Institute offered a one million dollar prize. some ideas that had intensely interested Gauss and Dirichlet. The resulting hastily-written, eight-page paper The Book introduced the complex-valued function now known as The authors begin by sorting their readership into three the Riemann zeta function and related it to the number groups: 1) interested individuals who may not have had a 휋(푋) of prime numbers less than or equal to a given calculus course, but who are comfortable with the notion quantity 푋. It introduced a host of astonishing new ideas of a graph of a function; 2) interested individuals who and techniques. As Selberg [2] points out, the paper was know some differential calculus; 3) interested individu- clearly a research note sketching some ideas to which Rie- als who know a little complex analysis. They then order mann intended to return. There were holes that needed their material accordingly. Part I, about seventy pages of to be filled and Riemann’s mathematical notebooks sub- the book, addresses the first group. Parts II and III, about sequently made it clear that Riemann had more material forty pages in total, address the second group. Part IV, the to present. Riemann would die seven years later, without shortest at twenty-five pages, addresses the third group, returning to the subject. At the time, the paper had lit- as do the ten pages of endnotes (which are terrific). tle discernible impact, but in time it would completely The first part opens with a careful discussion of primes, transform analytic number theory and reverberate across sieves, the frequency of primes, and the behavior of 휋(푋) mathematics. as 푋 grows larger. The prose is spare, but engaging and Thirty-five years later, Riemann’s paper was better un- informal. In addition to photographs of individuals (re- derstood. The Prime Number Theorem, originally conjec- grettably all male, all but one white), and excerpts from tured by Gauss and equivalent to the statement that no notebooks (see Figure 3), there are many tables and lots zeroes of the zeta function have real part equal to 1, was of well-chosen graphs (most generated by SageMath, with established independently by Hadamard and de la Vallée code available online). The function Li(푋) is simply de- Poussin in 1896. But a statement, now known as the Rie- fined as the area from 2 to 푋 under the graph of 1/ log(푥). mann Hypothesis, that Riemann conjectured would give No fuss, no over-explanation, no integral signs. The au- much better information, remains unsolved to this day. thors zoom out of the graph of the step function 휋(푋), Hilbert included it as the eighth problem among his list looking at it over larger and larger intervals. The reader of twenty-three problems for the twentieth century. After clearly sees the graph that becomes a smooth curve at the giving a nod to then-recent progress on the distribution 푋 resolution of the human eye and that the graph of log(푋) of prime numbers, he writes [2, p. 456] appears to diverge from 휋(푋) faster than Li(푋). It still remains to prove the correctness of an ex- Tables are used to show the distinction between going ceedingly important statement of Riemann, viz., to infinity at the same rate (as many leading digits as de- that the zero points of the function 휁(푠) defined sired can be made to coincide) and to within square root by the series error (first half of leading digits coincide). A wonderful 1 1 1 discussion that links square root error to random walks 휁(푠) = 1 + + + + … follows. This is done by simulating and plotting lots of 2푠 3푠 4푠 random walks on the line (see Figure 1), and it firmly es- 1 all have the real part 2 , except the well-known tablishes that square root error is the gold standard: the negative integral zeros. As soon as this proof has best that one can expect. been successfully established, the next problem The authors present their first formulation of the Rie- would consist in testing more exactly Riemann’s mann Hypothesis as the statement that Gauss’s logarith- infinite series for the number of primes below a mic integral Li(푋) is a square-root-close approximation given number and, especially, to decide whether to 휋(푋). This, of course, is the second part of Hilbert’s the difference between the number of primes statement of the Riemann Hypothesis, and their discus- below a number 푥 and the integral logarithm sion allows the reader to appreciate the contrast between of 푥 does in fact become infinite of an order the Prime Number Theorem and the Riemann Hypothesis. 1 not greater than 2 in 푥. Furthermore, we should From here, the account pivots to a discussion of the in- determine whether the occasional condensation formation contained in 휋(푋), and in the graphs of related of prime numbers which has been noticed in functions that contain the same information, which the counting primes is really due to those terms in authors refer to as carpentry on the staircase of primes. Riemann’s formula which depend upon the first In particular, the authors consider the step function 휓(푋) complex zeroes of the function 휁(푠). that increases not just at prime numbers by 1, but at all In the late 1990s, the Clay Mathematical Institute prime powers by the logarithm of the prime, and provide surveyed a representative group of mathematicians as a second restatement of the Riemann Hypothesis as the to what they considered the most pressing open prob- assertion that 휓(푋) is square-root-close to 푦 = 푋, the lems for the new millennium. There were, of course, graph of which is the straight line in the first quadrant at differences of opinion, but all approached included two 45 degrees off the 푋-axis. problems on their lists, the Poincaré Conjecture (now The discussion about preservation of information and solved) and the Riemann Hypothesis. Both were among the utility of different representations of functions leads 812 Notices of the AMS Volume 65, Number 7 BOOK REVIEW (A) Ten random walks (B) One hundred random walks (C) One thousand random walks Figure 1.