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

I. The 1859 paper “On the Number of Prime Numbers Less Than a Given Quantity” was Bernhard Riemann’s only publication on number theory, and the only one of his productions that contained no geometrical ideas at all.

The paper, though dazzling and seminal, was in some respects unsatisfactory. There was, first of all, the great Hypothesis, which Riemann left hanging in the air (where it still hangs). His actual words, after making a statement that is equivalent to the Hypothesis, were

Fair enough. Since the Hypothesis was not crucial to the ideas he was pursuing, Riemann left it unproved. That, however, is the least of the paper’s deficiencies. Several other things are asserted but not thor-

oughly proved—including the paper’s main result! (I shall give the result in a later chapter.)

Bernhard Riemann was a very pure case of the intuitive mathematician. This needs some explaining. The mathematical personality has two large components, the logical and the intuitive. Both are present in any good mathematician, but often one or the other is strongly dominant. The usual example of an extremely logical mathematician is the German analyst Karl Weierstrass (1815–1897), who did his great work in the third quarter of the nineteenth century. Reading Weierstrass’s papers is like watching a rock climber. Every step is firmly anchored in proof before the next step is taken. Poincaré said that none of Weierstrass’s books contained any diagrams. There is, in fact, just one exception to that, but certainly the precise logical progression of Weierstrass’s work, with every least fact carefully justified before proceeding to the next, and no appeals to geometrical intuition at all, is representative of the logical mathematician.

Riemann is at the other pole. If Weierstrass is a rock climber, inching his way methodically up the cliff face, Riemann is a trapeze artist, launching himself boldly into space in the confidence—which to the observer often seems dangerously misplaced—that when he arrives at his destination in the middle of the sky, there will be something there for him to grab. It is plain that Riemann had a strongly visual imagination, and also that his mind leaped to results so powerful, elegant, and fruitful that he could not always force himself to pause to prove them. He was keenly interested in philosophy and physics, and notions gathered from long, deep contemplation of those two disciplines—the flow of sensations through our senses, the organizing of those sensations into forms and concepts, the flow of electricity through a conductor, the movements of liquids and gases— can be glimpsed beneath the surface of his mathematics.

The 1859 paper is therefore revered not for its logical purity, and certainly not for its clarity, but for the sheer originality of the methods Riemann used, and for the great scope and power of his results,

which have provided, and will yet provide, Riemann’s fellow mathematicians with decades of research.

In his book on the zeta function,⁵⁰ Harold Edwards has this to say about what followed that 1859 paper.

II. Riemann’s “On the Number of Prime Numbers Less Than a Given Quantity” had a direct bearing on efforts to prove the Prime Number Theorem (PNT). If the Riemann Hypothesis were true, the PNT would follow as a consequence. However, the Hypothesis is a much stronger result than the PNT, and the latter could be proved from weaker premises. The main significance of Riemann’s paper for the proof of the PNT is that it provided the tools—the deep insights into analytic number theory that showed the way to a proof.

That proof came in 1896. The landmarks between Riemann’s paper and the proof of the PNT were as follows.

• There was an increase in the practical knowledge of prime numbers. Longer tables of primes were published, notably Kulik’s, deposited at the Vienna Academy in 1867, which provided factors for all numbers up to 100,330,200. Ernst Meissel developed a clever way to work out π (x), the prime counting function. In 1871 he produced a correct value for π (100,000,000). In 1885 he computed a value for

π (1,000,000,000), which was short by 56 (though this was not squareovered until 70 years later).

• In 1874, Franz Mertens proved a modest result about the series of reciprocals of primes, using methods that owed something to both Riemann and Chebyshev. That series, by the way, the series , diverges, though even more slowly than the harmonic series. It is ~log(log p).

• In 1881, J.J. Sylvester at Johns Hopkins University in the United States improved Chebyshev’s limits (see Chapter 8.iii) from 10 percent to 4 percent.

• In 1884 the Danish mathematician Jørgen Gram published a paper titled “Investigations of the Number of Primes Less Than a Given Number” and won a prize for it from a Danish mathematical society. (The paper made no important advances but laid the groundwork for Gram’s later efforts, which we shall examine in due course.)

• In 1885 the Dutch mathematician Thomas Stieltjes claimed to have a proof of the Riemann Hypothesis. More on this shortly.

• In 1890 the French Académie des Sciences announced that a grand prize would be awarded for a paper on the topic “Determination of the number of prime numbers less than a given quantity.” The deadline for presentation was June 1892. It was made plain in the announcement that the Académie was soliciting work that would supply some of the proofs missing from Riemann’s 1859 paper. The young French mathematician Jacques Hadamard submitted a paper concerning the representation of certain kinds of functions in terms of their zeros. Riemann had relied on this result to get a formula for π (x); it is on this point—I shall explain the math in more detail later— that the connection between prime numbers and the zeros of the zeta function hinges. Riemann, however, had left it unproved. The key ideas in Hadamard’s paper were drawn from

his own doctoral thesis, which he defended that same year. He won the prize.

• In 1895 the German mathematician Hans von Mangoldt proved the main result of Riemann’s paper, which states the connection between π (x) and the zeta function, and recast it in a simpler form. It was then plain that if a certain theorem much weaker than the Riemann Hypothesis could be proved, the application of the result to von Mangoldt’s formula would prove the PNT.

• In 1896 two mathematicians working independently, the aforementioned Jacques Hadamard and the Belgian Charles de la Vallée Poussin, proved that weaker result and, therefore, the PNT.

It had been said that whoever proved the PNT would attain immortality. This prediction very nearly came true. Charles de la Vallée Poussin died five months short of his 96th birthday; Jacques Hadamard two months short of his 98th.⁵¹ They did not know—not until late in the proceedings, anyway—that they were in competition with each other; and since both published in the same year, mathematicians consider it invidious to credit either with having got the result first. As with the ascent of Everest, the honor is shared.

In fact, de la Vallée Poussin seems to have been slightly earlier to press. Hadamard’s paper—its title was Sur la distribution des zéros de la fonction ζ (s) et ses conséquences arithmétiques—appeared in the bulletin of the Mathematical Society of France. Hadamard appended a note saying that while going over the galley proofs of the paper he had learned of de la Vallée Poussin’s result. He adds, “However, I believe no one will deny that my method has the advantage of simplicity.”

Nobody ever has denied it. Hadamard’s proof is simpler; and his knowing this before his paper went to press implies that he had not only heard of de la Vallée Poussin’s result but had had the chance to

examine it. However, since the two men’s work was plainly independent, and since there has never been any slightest suspicion of hanky-panky, and since both Hadamard and de la Vallée Poussin were perfect gentlemen, these simultaneous proofs have never generated any rancor or controversy. I am content to say, along with the whole world of mathematics, that in 1896, Jacques Hadamard of France and Charles de la Vallée Poussin of Belgium, working independently, proved the PNT.

III. The proving of the PNT is a great turning point in our story, so much so that I have divided my book into two parts on this point. In the first place, both of the 1896 proofs depended on getting a Hypothesis-style result. If either Hadamard or de la Vallée Poussin could have proved the truth of the Hypothesis, the PNT would have followed at once. They couldn’t of course, but they didn’t need to. If the PNT is a nut, the Riemann Hypothesis is a sledge hammer. The PNT follows from a much weaker result (which has no name):

All non-trivial zeros of the zeta function have real part less than one.

If you can prove this, then you can use von Mangoldt’s 1895 version of Riemann’s main result to prove the PNT. That is what our two scholars did in 1896.

In the second place, with the PNT out of the way, the Hypothesis came into plain view. It was the next great open issue in analytic number theory; and as mathematicians turned their attention to it, it soon became plain that if the Hypothesis could be shown to be true, a great many things would follow. If the PNT was the great white whale of number theory in the nineteenth century, the Riemann Hypothesis was to take its place in the twentieth. More than take its place, in

fact, for it cast its fascination not only on number theorists, but on mathematicians of all kinds, and even, as we shall see, on physicists and philosophers.

And in the third place—apparently trivial, but these things have a way of fixing themselves in people’s minds—there is the neat coincidence of the PNT being first thought of at the end of one century (Gauss, 1792), then being proved at the end of the next (Hadamard and de la Vallée Poussin, 1896). Once that theorem had been disposed of, the attention of mathematicians turned to the Riemann Hypothesis, which occupied them for the following century—which came to its end without any proof being arrived at. And that led inquisitive generalists to write books about the PNT and the Hypothesis at the beginning of the next century!

I am going to fill out the social, historical, and mathematical background to the bullet points given above by offering a sketch of the career of Jacques Hadamard; partly because he was the most important of the various players, and partly because I find him an appealing and sympathetic personality.

IV. Politically, France did not have a good nineteenth century. If Napoleon’s “100 days” are included (and if you will excuse a small rounding error), the constitutional arrangements of that ancient nation from 1800 to 1899 went as follows:

First Republic (4½ years)

First Empire (10 years)

Kingdom restored (1 year)

Empire restored (3 months)

Kingdom re-restored (33 years)

Second Republic (5 years)

Second Empire (18 years)

Third Republic (29 years)

… and even that 33 years of monarchy was interrupted halfway through by a revolution and change of dynasty.

For French people of the later part of the century, the great national trauma was the defeat of their armies by Prussia in 1870, followed by the Prussian siege of Paris in the winter of 1870–1871, then by a peace treaty that involved the cession of two provinces and a huge cash indemnity. The treaty itself triggered a brief but vicious civil war. The consequences of all this for France were, of course, very great. The nation went into the Franco-Prussian War an empire and came out a republic.

The French army was particularly affected. For the rest of the century and beyond, that proud institution not only had to bear the humiliation of the 1870 defeats; it also had to embody the hopes of the nation for revenge and for recovery of the lost territories. The army also became a focus for old-fashioned French patriotism, with young men from aristocratic, clerical, and high-bourgeois families joining the officer corps in large numbers. This tipped the officer class toward the old “Throne and Altar” style of French conservatism, and to some degree cut it off from the mainstream of French life in these decades. The mainstream was all in the direction of a bustling, open-minded, commercial, and industrial republic, a leader in the arts and sciences, a center of brilliance, wit, and gaiety—the wonderful, glittering France of the Belle Epoque, one of the great high points of western civilization.

Jacques Hadamard lived through the siege of Paris as an infant, and the house his family occupied was burned down in the Civil War. He had been born in December 1865 to French-Jewish parents. His father was a high-school teacher, his mother gave piano lessons. (Among her pupils was Paul Dukas, who wrote that Sorcerer’s Apprentice symphonic poem so well known to Disney fans.) After a degree and a brief spell of school teaching, Hadamard got his doctorate in 1892. He married that same year. In 1893 he moved with his wife to Bordeaux, where he took a position as lecturer at the university. The Hadamards’ first child, Pierre, was born in October 1894, and

they began raising one of those close, loving, busy bourgeois families in which everyone was expected to play a musical instrument and to enter business, academia, or the professions.

France was, then as today, a highly centralized nation. To get a lecturing position in Paris was extraordinarily difficult, and it was understood that young academics should serve an apprenticeship in the provinces for a few years. Hadamard’s Paris opportunity came in 1897. He moved back to the capital in that year, quitting his professorship in Bordeaux—he had advanced from lecturer to full professor in just two years—to become an assistant lecturer at the Collège de France—from the point of view of academic prestige, a move upward.

Those six years 1892–1897 laid the foundations of Hadamard’s career and fame. He was a mathematician of considerable scope, producing original work in several different areas. Undergraduate students of math generally first encounter his name attached to the Three Circles Theorem in complex function theory, a result Hadamard obtained in 1896, and which you can look up in any good encyclopedia of mathematics.⁵²

You will see it written that Hadamard was the last of the universal mathematicians—the last, that is, to encompass the whole of the subject, before it became so large that this was impossible. However, you will also see this said of Hilbert, Poincaré, Klein, and perhaps of one or two other mathematicians of the period. I don’t know to whom the title most properly belongs, though I suspect the answer is actually Gauss.

V. It is to the Bordeaux period that Hadamard’s proof of the PNT belongs. Permit me to step back a little and look at the immediate mathematical environment of the proof.

The senior figure in French mathematics at this time was Charles Hermite (1822–1901), professor of analysis at the Sorbonne until he

retired in 1897. One of his creations will play an important part later in our story (Chapter 17.v).

From 1882 onward Hermite had been conducting a mathematical correspondence with a younger mathematician, a Dutchman named Thomas Stieltjes.⁵³ In 1885, Stieltjes published a note in the Comptes Rendus⁵⁴ of the Paris Academy of Sciences, claiming to have proved my Theorem 15-1—a result stronger than the Riemann Hypothesis, from which, if Stieltjes had indeed proved it, the truth of the Hypothesis would follow (but whose falsehood would not disprove the Hypothesis—see Chapter 15.v). Stieltjes did not, however, include his proof in that note. He wrote to Hermite at about the same time, making the same claim, but adding, “My proof is very arduous; I shall try to simplify it further when I resume my research on these questions.” Now, Stieltjes was an honest man and a serious and respected mathematician—there is a type of integral named after him. No one had any reason to doubt that he did, in fact, have a proof, and in all probability Stieltjes himself thought he did.

Meanwhile, Riemann’s 1859 paper was being scrutinized, and its arguments tidied up. Hadamard’s prize result of 1892 was a great step forward. Then, in 1895 in Berlin, the German (Germany was by this time an Empire under Kaiser Wilhelm I) mathematician Hans von Mangoldt cleared away most of the remaining underbrush and proved Riemann’s main result linking the prime counting function π (x) to the zeros of the zeta function.

Only two large points remained, the Hypothesis and the PNT. By this time everyone concerned understood that the Hypothesis was the stronger proposition. If the Hypothesis (sledgehammer) could be proved true, the PNT (nut) would follow as a consequence, with no need for further effort; but the PNT could be established from weaker results without invoking the Hypothesis, and a proof of the PNT would not imply the truth of the Hypothesis.

So, what was a mathematician to do, given that it was widely believed that Stieltjes had disposed of both matters? Start work on proving the lesser result—to which, thanks to the brush-clearing work of

Hadamard and von Mangoldt, the way was now pretty clear? Was it worth the trouble, considering that Stieltjes’s superior result on the Hypothesis might appear while your own work was still in progress? On the other hand, by the mid-1890s it had been 10 years since Stieltjes’s announcement, and a lot of people must have been entertaining doubts. Not doubts about Stieltjes’s character; it is a very common thing for a mathematician to believe he has proved a result, only to find, going over his arguments (or more commonly, having them peer-reviewed), that there is a logical flaw in them. This happened with Andrew Wiles’s first proof of Fermat’s Last Theorem in 1993. It happens somewhat more dramatically to the narrator of Philibert Schogt’s 2000 novel The Wild Numbers. Nobody would have thought the worse of Stieltjes if this had been the case, this being much too common an event in mathematical careers. But where was that proof?

Both Charles de la Vallée Poussin at the University of Louvain in Belgium and Jacques Hadamard in Bordeaux took up the lesser challenge and soon got the result. They proved the PNT. Both must have wondered, though, whether there was any point to their efforts, since, even if their papers were to be published before Stieltjes’s, their lesser results would be overshadowed by his much greater one. Hadamard actually states in his paper: “Stieltjes has proved that all the imaginary zeros of ζ (s) are (conforming to Riemann’s prediction) of the form , t being real; but his proof has never been published. I simply intend to show that ζ (s) cannot have zeros with real part equal to 1.”

Stieltjes’s proof never did appear; and, in fact, Stieltjes had died in Toulouse on the last day of 1894. This fact must surely have been known to Hadamard, working on his paper in 1895–1896, so presumably he was expecting the proof to turn up in unpublished papers among Stieltjes’s effects. It never has. Until quite recently it was thought possible that Stieltjes might, nonetheless, have proved the Hypothesis. Then, in 1985, Andrew Odlyzko and Herman te Riele proved a result that casts serious doubt on Theorem 15-1. Belief in Stieltjes’s lost proof of Riemann’s Hypothesis has now, I think, pretty much evaporated.

VI. One consequence of the great national trauma of 1870–1871 was, as I have pointed out above, the reinforcing of the element of social conservatism in the officer class of the French army, and a certain distancing of that class from the main current of French society. This had one enormous consequence in the last years of the nineteenth century, the Dreyfus affair.

To attempt to do justice to “The Affair” in a couple of paragraphs is hopeless. It was a central issue in French public life for over a decade and can ignite a shouting match even today. There is a vast literature on it, along with movies, novels, and at least one TV mini-series (in French). As briefly as it can be stated: Alfred Dreyfus, an officer on the general staff of the French Army, from a wealthy Jewish-bourgeois family, was arrested and charged with treason at the end of 1894. Court-martialed in camera, he was condemned, degraded, and transported for life to Devil’s Island. Dreyfus loudly protested his innocence and had no apparent motive for treason, having always been impeccably patriotic and without any need for money.

In March 1896 Colonel George Picquart of French military intelligence happened to notice that the document that had been the principal item of evidence against Dreyfus was in fact in the handwriting not of Dreyfus but of another officer, Major Esterhazy, a man of erratic character and extravagant habits, chronically beset by gambling debts. Picquart informed his superiors. He was told to say nothing further about the matter and transferred to a frontier post in French North Africa. The following year, 1897, Dreyfus’s brother Mathieu learned of Picquart’s discovery and demanded that Esterhazy be tried. Esterhazy was acquitted by a military tribunal in January 1898. The novelist Émile Zola promptly published an open letter, the famous J’accuse, to the President of the Republic, Félix Faure, denouncing the various people involved in Dreyfus’s conviction as participants in a monstrous injustice and cover-up. Zola was indicted on a charge of criminal libel against the Minister of War.

The Affair then metastasized, consuming the attention of French society until Dreyfus’s innocence was finally and officially proclaimed

in July 1906. There were impassioned trials, dramatic reversals, the suicide of one of the conspirators, and numerous other colorful events. (Perhaps the most colorful, not arising directly from the Affair but influencing its course, was the death of President Faure while in flagrante delicto with his mistress in a back bedroom at the Elysée Palace. He suffered a massive stroke and in his death agony seized the poor woman by the hair with such force she was unable to separate herself from him. Her screams brought the Palace servants, who disengaged the lady, dressed her, and hustled her out a side door.)

It happened that Jacques Hadamard was a second cousin of Alfred Dreyfus’s wife Lucie, née Lucie Hadamard. The Affair was, therefore, of direct personal concern to him. In addition to this personal connection, it confronted all French Jews with deep questions about identity and loyalty. Before the Affair, most of the Jewish-French bourgeoisie—people like the Hadamards and the Dreyfuses— had thought themselves perfectly assimilated—patriotic French people who happened to be Jewish. Anti-Semitism had been lurking below the surface, however, and not only in the army. An anti-Semitic polemical book, La France Juive, had been a huge publishing success in 1886, and an anti-Semitic newspaper, La Libre Parole, was widely read. The Affair brought all this to the surface and made French Jews wonder if they had been living in a fool’s paradise. But even without the anti-Semitism factor a gross injustice had been done, and the ranks of the Dreyfusards—those agitating on behalf of the disgraced captain—included countless gentile citizens outraged by the army’s deceit and the failure of the political authorities to act.

Before the Affair, Hadamard seems to have been an apolitical and unworldly man, rather the absent-minded professor type that is very common among great mathematical talents. Much is made of this stereotype, and there is in fact something to it. Because of the purely abstract nature of the material they work with and the need to concentrate on that material for long hours at a time, mathematicians tend to be somewhat detached from more earthly matters. It is not impossible for a mathematician to be worldly, and there are many

counterexamples. René Descartes was a soldier and a courtier. (He survived the first but not the second.) Karl Weierstrass spent his years at university drinking and fighting and left without a degree. John von Neumann, one of the greatest of twentieth-century mathematicians, was quite a boulevardier, fond of pretty women and fast cars.

Jacques Hadamard, on the evidence, was not one of those counterexamples. Even discounting the apocrypha that develop around any great man, it seems plain that Hadamard could not knot his tie without assistance. His daughter claimed he could not count beyond four, “After that came n.” His involvement in the Dreyfus Affair, therefore, speaks to the depths of the feeling aroused by that incident, stirring even such a detached soul as this. Once he had become involved, Hadamard was a passionate Dreyfusard. He became active in the League for Human Rights, founded in 1898 during the trial of Zola. Hadamard’s third son, born in February 1899, was named Mathieu-Georges, “Mathieu” after Dreyfus’s brother and most tireless champion, “Georges” after the remarkable Colonel Picquart, whose iron integrity and quiet insistence on telling the truth were key factors in the eventual vindication of Dreyfus (whom Picquart personally detested).

Hadamard remained a public man for the rest of his life, which, as well as being exceptionally long, was more than usually productive and busy. It was also deeply marked with tragedy. The great wars of the twentieth century took all three of his sons. The older two died at Verdun within three months of each other; Mathieu-Georges was killed in 1944 while serving with the Free French forces in North Africa. In grief and despair after the First World War, Hadamard turned to pacifism and the League of Nations. He worked to help elect the Popular Front government of 1936–1938. Like many more worldly than himself, he was to some degree taken in by communism and the Soviet Union.⁵⁵ Driven from Paris by the German advance in 1940, he taught at Columbia for four years. He traveled and lectured everywhere and met everyone. He was a keen naturalist, with museum-grade collections of ferns and fungi. He was an early supporter of the

Hebrew University of Jerusalem (founded in 1925). His many books included The Psychology of Invention in the Mathematical Field (1945), still well worth reading for its insights into the thought processes of mathematicians; I have used some of its ideas for this book. He organized an amateur orchestra at his home; Albert Einstein—a lifelong friend—was a visiting violinist. He was married for 68 years to the same woman. When she died, Jacques was 94 years old. He struggled on for two years; but then the death of his beloved grandson in a climbing accident robbed him of his spirit and he died a few months later, a little short of his 98th birthday.

VII. In concentrating on Jacques Hadamard, I have indulged my personal fondness for an attractive personality and fine mathematical talent, intending no disrespect to the other mathematicians who participated in the clarification of Riemann’s great paper and the proof of the PNT.⁵⁶ By the later nineteenth century the world of mathematics had passed out of the era when really great strides could be made by a single mind working alone. Mathematics had become a collegial enterprise in which the work of even the most brilliant scholars was built upon, and nourished by, that of living colleagues.

One recognition of this fact was the establishment of periodic International Congresses of Mathematicians. The first such gathering was held in Zürich in August 1897. (Hadamard’s wife was expecting their second child, so he did not attend. He sent a paper to be read by his friend Emile Picard. It is interesting to note that the First Zionist Congress was taking place at the same time, 40 miles away in Basel, and inspired in part by issues arising out of the Dreyfus Affair.)

There was a second Congress in Paris in the summer of 1900, and the idea was to have a Congress every four years. History had other plans, however. There was no Congress in 1916, nor in 1940, 1944, or 1948. The system started up again in 1950 in Cambridge, Massachusetts. Hadamard was, of course, invited; but because of his pro-Soviet

leanings, he was at first denied a U.S. visa. It took a petition by his fellow mathematicians, and the personal intervention of President Truman, to get him to Harvard. At the time of writing, early in 2002, preparations are under way for the 24th Congress, to be held in Beijing this summer, only the second outside the West (defined as Europe, Russia, and North America).

VIII. The first of the twentieth-century Congresses was that one held in Paris from August 6 to 12, 1900, and this is the one everyone remembers. The Paris Congress will forever be linked with the name of David Hilbert, a German mathematician working at Göttingen, the university of Gauss, Dirichlet, and Riemann. Though only 38 years old, Hilbert was well established as one of the foremost mathematicians of his time.

On the morning of August 8, in a lecture hall at the Sorbonne, Hilbert stood up before the 200-odd delegates to the Congress, Jacques Hadamard among them, and delivered an address on “Mathematical Problems.” His aim was to concentrate the minds of his fellow mathematicians on the challenges facing them in the new century. To effect this goal, he directed their attention to a handful of the most important topics needing investigation, and problems needing solution. He organized these topics and problems under 23 headings; number 8 was the Riemann Hypothesis.

With that address, twentieth-century mathematics began in earnest.