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

I. We do not know much about Bernhard Riemann. He left no record of his inner life, other than what can be deduced from his letters. His friend and contemporary, Richard Dedekind, was the only person close to him who wrote a detailed memoir; but that was a mere 17 pages and revealed little. What follows, therefore, cannot hope to capture Riemann, but I hope it will at least leave him more than a mere name in the reader’s mind. I have reduced his academic career to a brief sketch in this chapter. I shall describe it in much more detail in Chapter 8.

First, let me set the man in his time and place.

II. Supposing that their Revolution had left the French disorganized and ineffective, and disturbed by its republican and antimonarchical ideals, France’s enemies moved to take advantage of the situation. In 1792 a huge force of mainly Austrian and Prussian troops, but which included 15,000 emigré French, advanced on Paris. To their surprise,

the army of revolutionary France took a stand at the village of Valmy, engaging the invaders in an artillery duel fought in thick fog on September 20 of that year. Edward Creasy, in his classic Fifteen Decisive Battles of the World, calls this the Battle of Valmy. Germans call it the Cannonade of Valmy. By either name it is a convenient marker for the beginning of the succession of wars that occupied Europe for the next 23 years. The Napoleonic Wars is the usual name given to these events; though it would be logical, if the expression were not already spoken for, to put them all under the heading First World War, since they included engagements in both the Americas and the Far East. When it all ended at last, with a peace treaty worked out at the Congress of Vienna (June 8, 1815), Europe settled into a long period, almost a century, of relative peace.

Northwest Germany after 1815. Note that Hanover (the state) is in two pieces; both Hanover (the city) and Göttingen belong to it. Prussia is in two large pieces and some smaller ones; both Berlin and Cologne are Prussian cities. Brunswick is in three pieces.

One consequence of the treaty was a modest tidying up of the German peoples in Europe. Before the French Revolution a German-speaking European might have been a citizen of Hapsburg Austria (in which case he was probably a Catholic) or of the Kingdom of Prussia (making him more likely a Protestant) or of any one of three hundred-odd petty principalities scattered across the map of what we now call Germany. He might also have been a subject of the king of France, or of the king of Denmark, or a citizen of the Swiss Confederation. “Tidying up” is a relative term—there was enough untidiness left over to occasion several minor wars, and to contribute to the two great conflicts of the twentieth century. Austria still had her empire (which included great numbers of non-Germans: Hungarians, Slavs, Romanians, Czechs, and so on); Switzerland, Denmark, and France still included German speakers. It was a good start, though. The three hundred-odd entities that comprised eighteenth-century Germany were consolidated into 34 sovereign states and 4 free cities, and their cultural unity was recognized by the creation of a German Confederation.

The largest German states were still Austria and Prussia. Austria’s population was about 30 million, only 4 million of them German speakers. Prussia had about 15 million citizens, most of them German speakers. Bavaria was the only other German state with a population over 2 million. Only four others had more than a million: the kingdoms of Hanover, Saxony, and Württemberg, and the Grand Duchy of Baden.

Hanover was something of an oddity in that, although a kingdom, its king was hardly ever present. The reason for this was that, for complicated dynastic reasons, he was also king of England. The first four of what English people call the “Hanoverian kings” were all named George,¹ and the fourth was on the throne in 1826, when the central character in the story of the Riemann Hypothesis first appeared.

III. Georg Friedrich Bernhard Riemann was born on September 17, 1826, in the village of Breselenz in the eastern salient of the Kingdom of Hanover. This part of the kingdom is known as Wendland, “Wend” being an old German word for the Slavic-speaking peoples they encountered. Wendland was the furthest west reached by the great Slavic advance of the sixth century. The name “Breselenz” itself derives from the Slavic word for “birch-tree.” Slavic dialects and folklore survived into modern times—the philosopher Leibnitz (1646– 1716) promoted research into them—but from the late Middle Ages onward German immigrants moved into Wendland and by Riemann’s time the population was pretty solidly German.

Wendland was, and still is, something of a backwater. With only 110 inhabitants per square mile, it is the most thinly populated district in its modern region, Lower Saxony. There is little industry and few large towns. The mighty Elbe—it is about 250 yards wide here— flows just 7 miles from Breselenz and was the principal connection with the world beyond until modern times. In the nineteenth century sailing ships and barges carried timber and agricultural produce down to Hamburg from Central Europe, returning with coal and industrial goods. During the recent decades of division, the Wendland stretch of the Elbe was part of the border between East and West Germany, a fact that did nothing to help local development. It is a flat, dull countryside of farm, heath, marsh, and thin woodland, prone to flooding. There was a serious flood in 1830 that must have been the first great external event of Bernhard Riemann’s childhood.²

Riemann’s father, Friedrich Bernhard Riemann, was a Lutheran minister and a veteran of the wars against Napoleon. He was already middle-aged when he married Charlotte Ebell. Bernhard was their second child and seems to have been especially close to his older sister, Ida—he named his own daughter after her. Four more children followed, a boy and three girls. With today’s standard of living, which of course we take for granted, it is difficult to imagine the hardships that faced a country parson, well into his middle years, with a wife

and six children to support, in a poor and undeveloped region of a middling country in the early nineteenth century. Of the six Riemann children, only Ida lived a normal life span. The others all died young, probably in part from poor nutrition. Riemann’s mother, too, died young, before her children were grown.

Poverty aside, it needs an effort of imagination for us, living and working in a modern economy, to grasp the sheer difficulty of finding a job in those times and circumstances. Outside large cities the middle class barely existed. There was a scattering of merchants, parsons, schoolteachers, physicians, and government officials. Everyone else who did not own land was a craftsman, a domestic servant, or a peasant. The only respectable employment for women was as governesses; otherwise they relied on their husbands or male family members for support.

When Bernhard was still an infant, his father took up a new position as minister in Quickborn, a few miles from Breselenz, and closer to the great river. Quickborn is still, today, a sleepy village of timber-framed houses and mostly unpaved streets bordered by massive, ancient oak trees. This place, even smaller than Breselenz, remained the family home until the elder Riemann died in 1855. It was the center of Bernhard’s emotional world until he was almost 30 years old. He seems to have returned there at every opportunity to be amongst his family, the only surroundings in which he ever felt at ease.

In reading of Riemann’s life, therefore, one must set it all against a backdrop of this environment, the environment of his home and upbringing, which he cherished, and for which, when away from it, he yearned. The flat, damp countryside; the draughty house lit only by oil lamps and candles, ill-heated in winter and ill-ventilated in summer; long spells of sickness among siblings who themselves were never quite well (they seem all to have suffered from tuberculosis); the tiny and monotonous social round of a parson’s family in a remote village; the inadequate and unbalanced diet on the stodgy side of a stodgy national cuisine (“For a long time he suffered from

chronic constipation,” notes Neuenschwander³). How did they stand it? But they knew nothing else and simple affection is sufficient to sustain the human spirit amid shared hardships.

IV. The multitude of states—kingdoms, principalities, duchies, and grand duchies—that made up North Germany in Riemann’s time were largely independent of each other and each made its own internal policy. This loose structure generated local pride and competition between the states.

In most respects they took their lead from Prussia. The eastern part of that kingdom was the only German state to keep some measure of independence from Napoleon after the defeats of 1806–1807. Under the stimulus of that brooding threat, the Prussians concentrated on internal reforms, overhauling their system of secondary education in 1809–1810 under the direction of the philosopher, diplomat, and linguist Wilhelm von Humboldt. Von Humboldt (whose brother Alexander was a great explorer and natural scientist) was a classicist and an ivory-tower man, who once said, “Alles Neue ekelt mich an.”—“All that is new disgusts me.” Yet oddly, the reforms brought in by this stern reactionary eventually made the educational systems of the German states the most academically advanced in Europe.

At the heart of the system was the 10-year gymnasium school, the years in question being age 10 to 20. In its earliest form, the curriculum at these schools was divided as follows.

By contrast, it is reported (by Jonathan Gathorne-Hardy in The Public School Phenomenon) that the great English boys’ schools of 1840 allocated 75–80 percent of teaching time—40 hours a week—to classics.

Quickborn had no gymnasium and Riemann did not begin his proper schooling until age 14, four years into the gymnasium course. This was in Hanover, the kingdom’s capital city, 80 miles from Quickborn. The location was determined by the fact of his maternal grandmother’s living in Hanover so that Riemann’s family was spared boarding fees. Before attending this gymnasium Riemann was educated by his father with some assistance from a village schoolteacher named Schultz.

Riemann, aged 14, was terribly unhappy in Hanover, morbidly shy and homesick. His only extracurricular activity, so far as we know, was seeking out such presents as he could afford to buy for his parents and siblings, to send to them on their birthdays. The death of his grandmother in 1842 led to a slight improvement. Riemann was transferred to another gymnasium, this one in the town of Lüneburg. Dedekind has this to say about the new situation.

Riemann does not seem to have been a good scholar. He had the type of mind that could hold only those things it found interesting, mathematics mostly. Furthermore, he was a perfectionist to whom conscientiousness in producing flawless essay compositions was more important than timeliness in delivering them. To improve his work the school director arranged for him to board with a teacher of Hebrew called Seffer or Seyffer. Under the care of this gentleman

Riemann improved sufficiently that in 1846 he was admitted to the University of Göttingen as a student of theology. The idea was that he would follow his father into the ministry.

V. Göttingen was the only university within the sphere of the Hanover church so it was the logical choice. The name “Göttingen” will crop up all through this book, so a few words about the history of the university may not be out of order. Founded in 1734 by George II of England (who was also Elector of Hanover⁵), Göttingen quickly became one of the better German provincial universities, with more than 1,500 students registered in 1823.

The 1830s, however, were a troubled time. Political agitation by both students and faculty lowered attendance to less than 900 in 1834. Three years later matters came to a head, and Göttingen attained a moment of Europe-wide fame. King William IV of England and Hanover died in 1837 without legitimate issue and the English throne passed to his niece, Victoria. Hanover, however, subscribed to the Salic Law of the medieval Franks, according to which only a male could succeed to the throne. England and Hanover thereupon parted company. The new ruler of Hanover was Ernest Augustus, oldest surviving son of George III.

Ernest Augustus was a great reactionary. Almost his first act was to set aside the liberal constitution granted by William IV four years earlier. Seven eminent professors at Göttingen University refused to swear an oath to uphold the new constitution and were dismissed. Three of them were actually exiled from the kingdom. These dismissed scholars became known as “the Göttingen Seven” and were heroes to social and political reformers all over Europe.⁶ Among them were the two brothers Grimm of fairy-tale fame, who were academic philologists.

In the changes that followed the continent-wide upheavals of 1848, Hanover got a new liberal constitution. At least one of the

Göttingen Seven, the physicist Wilhelm Weber, was reinstated. The university soon recovered its luster, eventually to become a great seat of learning, as we shall see. When Bernhard Riemann arrived in 1846, though, these upward trends were still in the future. He found Göttingen University a subdued place, attendance not yet recovered from the ructions of nine years earlier.

Göttingen did, however, have one major attraction for the young Riemann. It was the home of Carl Friedrich Gauss, the greatest mathematician of his age, and possibly of any age.⁷

Gauss was already 69 years old when Riemann arrived at Göttingen. His best work was behind him and he did little lecturing, regarding it as an annoying waste of time. Still his presence must have impressed Riemann, who had already been bitten by the math bug. We know that Riemann attended Gauss’s lectures on linear algebra and those of Moritz Stern on the theory of equations. At some point during this year 1846–1847 Riemann must have confessed to his father that he was far more interested in math than in theology and his father, who seems to have been a kind parent, gave his consent to mathematics as a career. And so Bernhard Riemann became a mathematician.

VI. Of Riemann’s adult personality, very little has come down to us. The primary source is the short memoir by Dedekind that I mentioned at the beginning of this chapter. The memoir was written 10 years after Riemann’s death and was appended to the first edition of his Collected Works (but never, so far as I know, translated into English).⁸ I have depended heavily on it for this book, so that many of the statements here and in Chapter 8 should really be tagged “… according to Dedekind.” You must take this as understood. Though Dedekind might, of course, have been mistaken on points of fact, he was the closest thing Riemann had to a friend. He was an honest and upright man and I have never seen any suggestion that he was less

than scrupulously truthful about his subject, with a single understandable exception that I shall mention in a moment. Other sources are Riemann’s private letters, many of which have survived, and some random recorded comments by students and colleagues.

These accounts tell us the following.

• Riemann was an extremely shy man. He avoided human contact as far as possible and was ill at ease in company. His only close ties—and they were very close indeed—were with his family, and his only other ties of any sort were with other mathematicians. When not among his family at the vicarage in Quickborn he suffered from homesickness.

• He was very pious, in the German Protestant style. (Riemann was Lutheran.) His opinion was that the essence of religion is, to translate literally from Dedekind’s German, “Daily self-examination before the face of God.”

• He thought deeply about philosophy and saw all his mathematical work in a larger philosophical context.

• He was a hypochondriac, in both the old and new senses of the word. (It was formerly a synonym for “depressive.”) Dedekind avoids this word, apparently out of consideration for Riemann’s widow, who begged that Riemann’s hypochondria not be made known. Dedekind makes it plain, though, that Riemann was subject to spells of very deep unhappiness, especially after the death of his father, whom he worshiped. Riemann dealt with these episodes by losing himself in work.

• His health was never good and was destroyed by the long years of privation to which a poor man had to resign himself if he was to get an advanced education in that time and place.

It is tempting to find Riemann a rather sad and slightly pathetic character. And yet that would be to consider only the outward appearance and manner of the man. Within that diffident, withdrawn exterior was a mind of great brilliance and staggering boldness. How-

ever timid and listless he may have appeared to casual observers, Riemann’s mathematics has the fearless sweep and energy of one of Napoleon’s campaigns. His mathematical friends and colleagues knew this, of course, and revered him.

Riemann brings to my mind an episode from Somerset Maugham’s novel The Moon and Sixpence, inspired by the life of the painter Gauguin. Maugham’s hero, like Gauguin an artist, dies of leprosy in a hut on a Pacific island, whither he has fled to pursue his vision of art. Hearing that the man is dying, a local doctor goes to his hut. It is a poor construction, shabby and dilapidated. When the doctor steps inside, however, he is astonished to find the interior walls all painted from floor to ceiling with brilliant, mysterious pictures. As with that hut, so it was with Riemann. Outwardly he was pitiable; inwardly, he burned brighter than the sun.

VII. In the realm of higher education, Wilhelm von Humboldt’s reforms had as yet left a mark only in Berlin, the Prussian capital. The situation in other German universities was as described by Heinrich Weber in his introduction to Riemann’s Collected Works.

Indeed, the von Humboldt reforms had for a while a negative effect on German higher education. They caused a demand for an increased supply of well-trained secondary-school teachers, and the only way this demand could be met was for the universities to do the training. Even the mighty Gauss was teaching mainly elementary courses at Göttingen University in 1846–1847. In search of a meatier

diet, Riemann transferred to Berlin University. Two years at that institution, under instruction from the best mathematical minds in Germany, brought Riemann to full maturity as a mathematician.

(Here and throughout these early historical chapters, you should understand that in Europe before the post-Napoleonic shift of attitudes, and in some countries for longer, there was a clear distinction between universities, whose purpose was to teach and train whatever of a cognitive elite the nation was thought to require, and academies or societies, which existed for the purpose of research—this being understood, to a greater or lesser degree depending on the time, the place, and the inclination of the ruler, to be for the practical advantage of the state. Institutions like Berlin University, founded in 1810, where some research was done, and the early St. Petersburg Academy, where teaching went on, were rare exceptions to this general rule. The Berlin Academy, where the Riemann Hypothesis first saw the light of day, was a pure-research establishment modeled on England’s Royal Society.)

We know next to nothing about Riemann’s everyday life in Berlin outside his mathematical studies. Dedekind records only one incident worth noting. In March 1848 the Berlin mob, inspired by the February revolution in Paris, took to the streets, demanding the unification of the German states into a single empire. Barricades went up, the army tried to clear them, and blood was shed. The Prussian king at the time was Friedrich Wilhelm IV, a rather dreamy and unworldly man, much under the influence of the Romantic Movement, with a sentimental view of his people and an ideal of the state as a paternalistic monarchy. He proved maladroit in the crisis, sending the army back to camp and leaving his palace unprotected before the insurrectionists had been dispersed. The university students formed a loyal guards corps to protect the king and Riemann served a spell of guard duty with this corps from 9:00 one morning until 1:00 the following afternoon, a grand total of 28 hours.

After returning to Göttingen in 1849, Riemann began work for his doctorate, which he attained two years later, at age 25, having submitted a dissertation on complex function theory. He became a lec-

turer at Göttingen three years after that and an associate professor in 1857—his first salaried position. (Ordinary lecturers were expected to survive on fees paid by whatever students they could attract to their lectures. The job title was Privatdozent—“private lecturer.”)

The year 1857 was also what we should call, in the language of current celebrity biography, Riemann’s “breakout year.” His 1851 doctoral dissertation is nowadays regarded as a classic of nineteenth-century mathematics, but it drew little attention at the time in spite of having been enthused over by Gauss. His other written papers of the early 1850s were not widely known and were published in an accessible form only after his death. To the degree that he had become known at all, it was mainly through the content of his lectures; and much of that content was too far ahead of its time to be appreciated. In 1857, however, Riemann published a paper on analysis that was at once recognized to be a major contribution. Its title was “Theory of Abelian Functions.”⁹ In it, he tackled topical problems by ingenious and innovative methods. Within a year or two his name was known to mathematicians all over Europe. In 1859 he was promoted to full professor at Göttingen, at last attaining sufficient income to allow him to marry—which he did, three years later. His bride was Elise Koch, a friend of his oldest sister.

On August 11 of that same year, 1859, shortly before his 33rd birthday, Bernhard Riemann was also appointed a corresponding member of the Berlin Academy. The Academy based their decision on the only two of Riemann’s papers that were well known, the 1851 doctoral dissertation and the 1857 work on Abelian functions. To be elected a member of the Berlin Academy was a great honor for a young mathematician. It was the custom to acknowledge such appointments by submitting an original paper to the Academy, describing some research one was engaged in. The paper Riemann submitted was titled “On the Number of Prime Numbers Less Than a Given Quantity” (Über die Anzahl der Primzahlen unter einer gegebenen Grösse).

Mathematics has not been quite the same since.