Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics (2003)

I. Göttingen was not, of course, the only place where first-class mathematics was being done in the early years of the twentieth century. Here is the English mathematician John Edensor Littlewood, 60-odd years before he offered snuff to Hugh Montgomery. As a young mathematician at Trinity College, Cambridge, in 1907, Littlewood was casting around for a good meaty problem on which to do postgraduate research.

I have taken that passage from Littlewood’s Miscellany, a quirky collection of autobiographical fragments, jokes, math puzzles, and character sketches, first published (under a slightly different title) in 1953. The other dramatis personae in the extract are the older English mathematician Godfrey Harold Hardy, 1877–1947, and the German Edmund Landau, 1877–1938. These three men, half a generation after Hilbert, were all pioneers in the early assaults on the Riemann Hypothesis.

II. British mathematics in the nineteenth century had been oddly asymmetrical in its development and achievements. Great advances were made by British mathematicians in the least abstract areas of math, those most closely connected with physics. This was something I noticed during my own higher-mathematical education in London. We would sit through a class in real analysis, or complex function theory, or number theory, or algebra, and the names attached to the theorems would come rolling in across the English Channel from the Continent: Cauchy, Hadamard, Jacobi, Chebyshev, Riemann, Hermite, Banach, Hilbert…. Then we would have a Methods lecture (i.e., on mathematical methods used in physics), and suddenly we were back in Victoria’s islands: Green’s Theorem (1828), Stokes’s Formula (1842), the Reynolds Number (1883), Maxwell’s Equations (1855), the Hamiltonian (1834)….

Such other activity as took place in Britain was concentrated in the most abstract areas of math. Arthur Cayley, with J.J. Sylvester, invented matrices (more about them later), and the theory of algebraic invariants. George Boole opened up the whole territory of “foundations”—that is, mathematical logic, which he called “the laws of thought.” (You can get an argument going about whether this is really at the high end of the abstraction scale. Boole himself declared that his intention was to make logic a branch of applied mathematics. However, I think mathematical logic is sufficiently abstract for most of us mortals.) It is curious to note that the week before Hilbert addressed the Paris Congress, the same lecture rooms at the Sorbonne had been booked for an International Congress of Philosophy. One of the papers read was “The Idea of Order and Absolute Position in Space and Time.” Its author was a young British logician, also a Trinity man, named Bertrand Russell, who 10 years later, with Alfred North Whitehead, produced the classic of mathematical logic (to be more precise, of logicized mathematics), Principia Mathematica.

The least abstract math, and the most, but the great middle ground of abstraction—function theory, number theory, most of algebra—was yielded to the Continentals. In analysis, the most fertile field of nineteenth-century mathematics, the British were nearly invisible. At the end of the century they were in fact barely visible even in their strong areas. Only seven British mathematicians showed up at the Paris Congress, ranking Britain below France (90), Germany (25), the U.S.A. (17), Italy (15), Belgium (13), Russia (9), Austria, and Switzerland (8 each). Mathematically, Britain in 1900 was a backwater.

Even a backwater, of course, has some pockets of vitality. Trinity College, Cambridge, where Littlewood was in residence, maintained a strong mathematical tradition. It had been Sir Isaac Newton’s college, 1661–1693, and counted several geniuses of mathematics and physics among its nineteenth-century alumni: Charles Babbage, generally credited with inventing the computer; the astronomer George Airy, after whom a family of mathematical functions is named;

Augustus de Morgan, the logician; Arthur Cayley, the algebraist; James Clerk Maxwell, and some lesser lights. Bertrand Russell got his degree at Trinity in 1893, was elected a fellow⁷³ in 1895, and was teaching there at the time Hardy joined the faculty. The college’s history in the twentieth century was somewhat more mixed. It supplied most of the personnel for the Cambridge spy ring,⁷⁴ as well as several Bloomsberries.⁷⁵ So far as mathematics was concerned in the early years of the century, though, it was first and foremost the home of G.H. Hardy—the Hardy of Littlewood’s memoir. It was Hardy, more than anyone else, who awoke English pure mathematics from its long slumber.

Studying for his degree at Trinity in 1897, Hardy came across a famous textbook of the time, Cours d’Analyse, by the French mathematician Camille Jordan. Jordan is familiar to students of complex variable theory for Jordan’s Theorem, which says, basically, that a simple closed curve in the plane, for example a circle, has an inside and an outside. This theorem is ferociously difficult to prove— Estermann describes Jordan’s own proof as “an intelligent attempt.” Cours d’Analyse seems to have had the same effect on Hardy as Chapman’s Homer had on Keats. After getting his fellowship at Trinity in the summer of Hilbert’s address, Hardy spent the next few years publishing papers on analysis.

One fruit of Hardy’s early analytical obsession was an undergraduate textbook, A Course of Pure Mathematics, first published in 1908 and never subsequently out of print. I learned analysis from this book, as did most twentieth-century British undergraduates. We referred to the book simply as “Hardy.” The book’s title is entirely misleading, as it contains nothing but analysis—no algebra, no number theory, no geometry, no topology. Nobody has ever minded this, though. As an introduction to classical (i.e., nineteenth-century) analysis, it is as near to perfect as a textbook can be. Its influence on my own approach to math was tremendous. Looking through what I have written in this book, I see Hardy all too plainly.

III. G.H. Hardy is the kind of oddity that only nineteenth-century England could produce. In old age he wrote a very curious book titled A Mathematician’s Apology (1940), in which he described his own life as a mathematician. It is in some ways a sad book—an elegiac book, to be precise. The reason for this is very well explained in C.P. Snow’s preface to the later editions. Hardy was a Peter Pan, a boy who never grew up. Snow: “His life remained the life of a brilliant young man until he was old: so did his spirit: his games, his interests, kept the lightness of a young don’s. And, like many men who keep a young man’s interests into their sixties, his last years were the darker for it.” Littlewood: “Until he was about 30 he looked incredibly young.” Hardy’s games were cricket, about which he was passionate, and real tennis (a.k.a. court tennis or jeu de paume), a more difficult, more intellectually challenging game than ordinary tennis.

For 12 years, 1919–1931, Hardy held a chair at Oxford, with an exchange year at Princeton, 1928–1929; the rest of his life was spent at Trinity, Cambridge. A handsome and charming man, he never married, nor had any intimate attachments of any kind, so far as anyone knows. It must be remembered that the old Oxford and Cambridge colleges were men-only institutions with a strong flavor of misogyny. Until 1882, Fellows of Trinity were not permitted to marry. In the manner of our age, there has recently been some speculation that Hardy may have been homosexual. I refer the curious reader to Robert Kanigel’s biography of Hardy’s protégé Srinivasa Ramanujan, The Man Who Knew Infinity, which contains a full discussion of this point. The answer seems to be: probably not, except perhaps in the innermost sense.

There are even more Hardy stories than there are Hilbert stories—I see that I have already told one. Here are two more, both containing the Riemann Hypothesis. The first is from his obituary in the British science journal Nature.

1. prove the Riemann Hypothesis;

2. make 211 not out in the fourth innings of the last Test Match at the Oval;

3. find an argument for the non-existence of God which shall convince the general public;

4. be the first man at the top of Mount Everest;

5. be proclaimed the first president of the USSR of Great Britain and Germany;

6. murder Mussolini.

The second illustrates another of Hardy’s eccentricities. Though claiming not to believe in God, he carried on a perpetual battle of wits with Him. In the 1930s, Hardy often visited with his friend Harald Bohr, who was Professor of Mathematics at the University of Copenhagen (and younger brother of the physicist Niels Bohr). George Pólya told the following story about one of these trips.

His wonderful textbook aside, Hardy is best known for two great collaborations of which he was a part. The one with Ramanujan has been best publicized, and for good reason because it is one of the most curious and affecting stories in the history of mathematics. It is

told in full in the aforementioned book by Robert Kanigel. However, the Hardy-Ramanujan collaboration is of only the most incidental concern to the history of the Riemann Hypothesis, and I shall have no more to say about it.

Hardy’s other great collaboration was with Littlewood, with whose memoir about his own postgraduate research I opened this chapter. Littlewood joined the Trinity faculty in 1910. His collaboration with Hardy began the following year and continued until 1946. It was conducted mostly by mail during the years that Hardy was at Oxford and Princeton, and also during World War I, when Littlewood worked on artillery matters for the British army. Collaboration by mail was not much of a departure for Hardy and Littlewood, though: they often communicated by mail when living in rooms at Trinity.

Both Hardy and Littlewood were great mathematicians, both were the sons of schoolmasters, and both were lifelong bachelors. In most other ways they were different. There is something distinctly strange about Hardy. He hated having his photograph taken, for example—there are only half a dozen extant photographs of him⁷⁶— and when staying in a hotel or guest room, he would cover up all the mirrors. Littlewood was much more of a meat-and-potatoes man. Where Hardy was slender and finely made, Littlewood was stocky and strong, a good all-round sportsman: swimming, rowing, rock climbing, cricket. He took up skiing at age 39 and became very proficient—an unusual thing among Englishmen at that time. He loved music and dancing.

Though conforming to the old idea of a college fellow—never married, he occupied the same set of rooms at Trinity for 65 years, 1912–1977—Littlewood had at least two children. The story as his colleague Béla Bollobás tells it is that Littlewood, in his younger years, used to go for annual vacations with the family of a doctor in Cornwall, whose children grew up calling him “Uncle John.” One of these children was named Ann; Littlewood referred to her as “my niece.” However, after becoming close friends with Bollobás and his

wife, Littlewood confessed that Ann was, in fact, his daughter. They persuaded him to stop calling her his niece and start saying “my daughter.” He accordingly did so, in the faculty common room one evening, and was mortified that none of his colleagues displayed the least surprise. Then, after Littlewood’s death in 1977, a middle-aged man showed up at Trinity asking about his effects, explaining that he was Littlewood’s son.

IV. “Hardy and Littlewood” became such a common byline on mathematical papers in the 1910s and 1920s that jokes were circulating about Littlewood being a fiction, invented by Hardy to take the blame for his mistakes. One German mathematician was said to have crossed the English Channel solely to confirm his belief that Littlewood did not exist.

That mathematician was Edmund Landau, who was seven days younger than Hardy. Landau was an instance of that uncommon phenomenon, the scion of a wealthy family who yet had a powerful work ethic and a record of great achievement in a non-commercial field. Landau’s mother Johanna, née Jacoby, came from a rich banking family. His father was a Professor of Gynecology in Berlin, with a successful practice. Landau Senior was also a keen supporter of Jewish causes. The family home was at Pariser Platz 6a, in the most elegant quarter of Berlin, close to the Brandenburg Gate. Edmund was appointed to a professorship at Göttingen in 1909. When people asked for directions to his house, he would reply “You can’t miss it. It’s the finest house in town.” He followed his father’s (and Jacques Hadamard’s) interest in Zionism, helping to establish the Hebrew University of Jerusalem and giving the first math lecture there, in Hebrew, shortly after the university opened in April 1925.

Landau was something of a character—this was a great age for mathematical characters—and there are apocrypha about him rivaling those of Hilbert and Hardy. Perhaps the best-known story is his

remark about Emmy Noether, a colleague at Göttingen. Noether was mannish and very plain. Asked if she was not an instance of a great female mathematician, Landau replied: “I can testify that Emmy is a great mathematician, but that she is female, I cannot swear.” His work ethic was legendary. It is said that when one of his junior lecturers was in hospital, recuperating from a serious illness, Landau climbed a ladder and pushed a huge folder of work through the poor man’s window. Littlewood: “He simply did not know what it was like to be tired.” Hardy says that Landau worked from 7 A.M. until midnight every day.

Landau was a gifted and enthusiastic teacher, and an extraordinarily productive mathematician. He wrote more than 250 papers and 7 books. His main importance for our story is the first of those books, a classic of number theory, published in 1909. This is the book Littlewood was speaking of in the extract I opened this chapter with: “All this was transformed at a stroke by the appearance of Landau’s book….” The book’s full title was Handbuch der Lehre von der Verteilung der Primzahlen—“Handbook of the Theory of the Distribution of the Prime Numbers.” It is generally referred to by number theorists as simply “the Handbuch.” ⁷⁷ In two volumes of more than 500 pages each, this book gathered together all that was known about the distribution of primes up to that time, with a strong emphasis on analytic number theory. The Riemann Hypothesis is stated on page 33. The Handbuch was not the first book on analytic number theory—Paul Bachmann had published one in 1894—but its extremely detailed and systematic presentation laid out the subject in a style both clear and attractive, and Landau’s book at once became the standard in its field.

I don’t think Landau’s Handbuch has ever been translated into English. Number theorist Hugh Montgomery, the star of my Chapter 18, taught himself German by reading his way through the Handbuch, one finger on the dictionary. He tells the following story. The first 50-odd pages of the book are given over to a historical survey, in sections each of which is headed with the name of a great mathematician who

made contributions in the field: Euclid, Legendre, Dirichlet, and so on. The last four of these sections are headed “Hadamard,” “von Mangoldt,” “de la Vallée Poussin,” “Verfasser.” Hugh was extremely impressed with the contributions of Verfasser, but was puzzled to know why he had not heard the name of this fine mathematician before. It was some time before he learned that “Verfasser” is a German word meaning “author” (ordinary nouns are capitalized in German).

V. “All this was transformed at a stroke by the appearance of Landau’s book….” Both Hardy and Littlewood must have read Landau’s book soon after it became available. Here is what Hardy has to say, in the obituary of Landau he wrote (with Hans Heilbronn) for the London Mathematical Society.

It was certainly from the Handbuch that both Hardy and Littlewood became infected with the Riemann Hypothesis obsession. The first fruits came in 1914, not in the form of a collaboration, though they were collaborating by that time, but as two separate papers, both of major importance in the development of the theory.

Hardy’s paper was titled Sur les zéros de la fonction ζ (s) de Riemann and appeared in the Comptes Rendus of the Paris Academy of Sciences. In it, he proved the first major result on the distribution of the non-trivial zeros.

Hardy’s 1914 Result

Infinitely many of the zeta function’s non-trivial zeros satisfy the Riemann Hypothesis—that is, have real part one-half.

Though a major step forward, it is important for the reader to understand that this did not settle the Hypothesis. There is an infinity of non-trivial zeros; Hardy proved that infinitely many of them have real part one-half. This leaves three possibilities still open:

• Infinitely many zeros do not have real part one-half.

• Only finitely many zeros do not have real part one-half.

• There are no zeros that do not have real part one-half—the Hypothesis!

For an analogy, consider the following statements about the even numbers greater than two, that is: 4, 6, 8, 10, 12, ….

• Infinitely many of them are divisible by 3; infinitely many are not.

• Infinitely many are greater than 11; only four are not.

• Infinitely many are the sum of two primes; there are none that are not—the Goldbach Conjecture (which is still unproven at the time of writing).

Littlewood’s paper, also published in the Paris Academy’s Comptes Rendus of that year, was titled Sur la distribution des nombres premiers. It proved a result as subtle and striking as Hardy’s, though in a different part of the field. It needs some preamble.

VI. I have already pointed out the following general trend in thinking about the Riemann Hypothesis at the beginning of the twentieth century. The Prime Number Theorem (PNT) had been proved. It was

known with mathematical certainty that indeed π (x) ~ Li(x)—to put it in words, that the relative difference between π (x) and Li(x) dwindles away to zero as x gets bigger and bigger. So now what can we say about this difference, this error term? It was in focusing on the error term that mathematicians’ attention was drawn to the Riemann Hypothesis, because Riemann’s 1859 paper gave an exact expression for the error term. That expression, as I shall show in due course, involves all the non-trivial zeros of the zeta function, so the key to understanding the error term is hidden in among the zeros somehow.

Let me make this concrete by showing some actual values of the error term. In Table 14-1, “absolute error” means Li(x) – π (x), while “relative error” means that number as a proportion of π (x)—in other words, the absolute error divided by π (x).

Well, the relative error is certainly dwindling away to zero, just as the PNT says it should. This is happening because the absolute error, though increasing, is not increasing anything like as fast as π (x).

The inquiring mathematical mind now asks how, exactly, do these numbers behave? Are there rules to describe the slow increase of the absolute error, or the dwindling to zero of the relative error? To put it another way, if you drop the second and fourth columns of Table 14-1, or the second and third, and consider the resulting two-column tables to be snapshots of some functions (argument, value)—what

functions are they? Can we get twiddle formulas for them, as we did for π (x)?

That is where the non-trivial zeros of the zeta function come in. They are intimately connected, in a way I shall later show you in exact mathematical detail, with the error term.

Although it is the relative error that the PNT speaks about, investigations in this area more often concentrate on the absolute error. It really makes no difference, of course, which one you consider. The relative error is just the absolute error divided by π (x), so you can always skip easily from one to the other. So can we get any kind of result for the absolute error term, Li(x) – π (x)?

VII. Looking at Figure 7-6, and at Table 14-1, we can say with fair confidence that the the absolute difference Li(x) – π (x) is positive and increasing. The numerical evidence for this is so strong that Gauss, when he made his own investigations, believed it to be always the case. Probably most early researchers agreed, or at least felt sure that π (x) is always less than Li(x). (Riemann’s opinion on the matter is unclear.) Littlewood’s 1914 paper therefore came as a sensation, for it proved that this is not so; that, on the contrary, there are numbers x for which π (x) is greater than Li(x). It actually proved much more.

Littlewood’s 1914 Result

Li(x) – π (x) changes from positive to negative and back infinitely many times.

Given that π (x) is less than Li(x) for as far as we have been able to take x, even with the most powerful computers, where is that first crossing point, the first “Littlewood violation,” where π (x) becomes equal to, and then greater than, Li(x)?

In situations like this, mathematicians go looking for what they call an upper bound, that is, a number N for which they can prove that

whatever the precise answer to the question, it is at any rate definitely less than N. Proven upper bounds N of this sort are sometimes far larger than the actual answer.

That was the case with the first upper bound for the Littlewood violation. In 1933 Littlewood’s student Samuel Skewes showed that if the Riemann Hypothesis is true, the crossover point must come before , a number of about 10^{ten billion trillion trillion} digits. That’s not the number; that’s the number of digits in the number. (By way of contrast, the number of atoms in the cosmos is thought to have about eighty digits.) This monstrosity attained fame as “Skewes’ number,” the largest number ever to emerge naturally from a mathematical proof up to that time.⁷⁸

In 1955 Skewes improved his result, this time without assuming the truth of the Riemann Hypothesis, to a number of a mere 10^{one thousand} digits. In 1966, Sherman Lehman pulled the upper bound down to a much more manageable (or at least, writable) figure, 1.165 × 10¹¹⁶⁵ (a number, that is, of a mere 1,166 digits), and established an important general theorem about the upper bound. In 1987, using Lehman’s theorem, Herman te Riele reduced the upper bound still further, to 6.658 × 10³⁷⁰.

At the time of writing (mid-2002), the best figure is the one established by Carter Bays and Richard Hudson in 2000, also starting from Lehman’s theorem.⁷⁹ They showed that there are Littlewood violations in the vicinity of 1.39822 × 10³¹⁶ and even gave some reasons for thinking that these may be the first violations. (Bays’s and Hudson’s paper leaves open a small possibility that lower violations might exist, perhaps even as low as 10¹⁷⁶. They also show a huge zone of violation around 1.617 × 10⁹⁶⁰⁸.)

VIII. These oscillations of the error term Li(x) – π (x) from positive to negative and back take place within fairly well-defined constraints, though. If this were not so, the PNT would not be true. Some

ideas about the nature of those constraints had already emerged out of the effort to prove the PNT. De la Vallée Poussin had actually included an estimate for the constraining function in his own proof of the PNT. Five years later, in 1901, the Swedish mathematician Helge von Koch⁸⁰ had proved the following key result, which I’ll state in a modern form.

Von Koch’s 1901 Result

If the Riemann Hypothesis is true, then

The equation is pronounced as, “Pi of x equals log integral of x plus big oh of root x log x.” Now I have to explain the “big oh” notation.