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

I. The first person to whom the truth contained in the Prime Number Theorem (PNT) occurred was Carl Friedrich Gauss, whose dates were 1777 to 1855. Gauss has, as I mentioned in Chapter 2.v, a good claim to being the greatest mathematician who ever lived. In his lifetime he was known as Princeps Mathematicorum—the Prince of Mathematics—and at his death the King of Hanover, George V, ordered a commemorative medal in his honor, with that title on it.¹⁴

Gauss came from extremely humble origins. His grandfather was a landless peasant; his father was a jobbing gardener and bricklayer. Gauss attended the poorest kind of local school. A famous incident, reported from that school, is much more likely to be true than most such stories are. One day the schoolmaster, to give himself a half-hour break, set the class to adding up the first 100 numbers. Almost instantly, Gauss threw his slate onto the master’s table, saying, “Ligget se!” which in the peasant dialect of that place and time meant, “There it is!” Gauss had mentally listed the numbers horizontally in order (1, 2, 3, …, 100), then in reverse order (100, 99, 98, …, 1) then added the

two lists vertically (101, 101, 101, …, 101). That is 100 occurrences of 101, and since all the numbers were listed twice, the required answer is half this sum: 50 times 101, which is 5,050. Easy when you have been told it, but not a method that would occur to the average 10-year-old; nor even the average 30-year-old, for that matter.

It was Gauss’s good luck that his schoolmasters recognized his ability and were willing to go to some pains to promote it. It was his even greater luck to live in the small German duchy of Brunswick— the blob that separates the two parts of Hanover on the map in Chapter 2.ii. Brunswick was ruled at this time by Carl Wilhelm Ferdinand, who rejoiced in the title Herzog zu [that is to say, “Duke of ”] Braunschweig-Wolfenbüttel-Bevern. We have met this Duke already without knowing it at the time. A keen soldier all his life, he held the rank of field marshal in the Prussian army and was in charge of the joint Prussian-Austrian force that the French stopped at Valmy on September 20, 1792.

Carl Wilhelm truly was a gentleman. If there is a mathematicians’ Heaven, some sumptuous apartments must be set aside in it for him, for his use whenever he feels inclined to visit. Hearing of the boy Gauss’s talent, the Duke asked to see him. Young Gauss cannot have possessed much in the way of social polish at this point. Later in life, after much acquaintance with courts and universities, he is described as mild and affable; but he always had the rough-cut features and stocky physique of his peasant origins. However, the Duke was sufficiently discerning that he took to the boy at once, remained his friend until death parted them, and provided the steady financial support that enabled young Gauss to embark on a long brilliant career as a mathematician, physicist, and astronomer.¹⁵

The Duke’s ability to support Gauss ended very tragically. In 1806 Napoleon was at the height of his career. In the previous year’s campaigning, he had defeated the combined armies of Russia and Austria at the battle of Austerlitz, having temporarily bought off the Prussians by offering them Hanover. He had then established the Confederation of the Rhine, bringing all the western part of what is now Ger-

many under French rule, and reneged on the Hanover deal, offering it now to Britain. Only Prussia and Saxony held out against him; and their only ally was Russia, gun-shy from the defeat at Austerlitz.

To prevent Saxony from becoming a French satellite, the Prussians occupied it, calling the Duke of Brunswick out of retirement—he was 71 years old at this point—to lead their forces. Napoleon declared war and his army struck northwest through Saxony toward Berlin. The Prussians tried to concentrate forces, but the French were too fast for them, and crushed the main Prussian units at Jena. The Duke was with a detachment at Auerstädt a few miles to the north; one of Napoleon’s flanking corps caught him and routed his troops.

Defeated and mortally wounded, the Duke asked Napoleon, via an emissary, for leave to return to his home to die. The Emperor, a thoroughly modern dictator who was not much given to chivalry, laughed in the messenger’s face. The unfortunate Duke, blinded and dying, had to be hurried away in a cart to the free territories beyond the Elbe. Napoleon’s secretary, Louis de Bourienne, tells the melancholy end of the tale in his Memoirs.

He had passed through Brunswick on the way, and it is said that Gauss saw the cart from the window of his room opposite the castle gate. The Duchy of Brunswick was then wound up, incorporated into

Napoleon’s puppet “Kingdom of Westphalia.” The Duke’s heir, Friedrich Wilhelm, was dispossessed and had to flee to England. He, too, died fighting Napoleon, at the battle of Quatre Bras in 1815, a few days before Waterloo, but not before his duchy had been restored to him.

(In strict fairness to Napoleon, I should add that on a later razzia through western Germany, when Gauss was installed at Göttingen, the Emperor spared the city because “the greatest mathematician of all time is living there.”)

II. Having lost his patron, Gauss had to find a job. He was offered, and took, the position of director of the observatory at Göttingen University, arriving there in late 1807.¹⁶ Göttingen was already known as one of the better-equipped provincial German universities. Gauss had studied there himself in 1795–1798, apparently attracted by its splendid library, where he had spent most of his time. Now he became head of astronomy at the university and stayed at Göttingen until his death in February 1855, a few weeks short of his 78th birthday. In the last 27 years of his life, he slept away from his beloved observatory only once, to attend a conference in Berlin.

To tell of Gauss’s connection with the PNT, I must explain his chief peculiarity as a mathematician. Gauss published much less than he wrote. We know—from his correspondence, his surviving unpublished papers, and circumstantial evidence in his published works— that what he presented to the world was only part of what he discovered. Theorems and proofs that would have made another man’s reputation, Gauss left languishing in his personal diaries.

There seem to have been two reasons for this apparent carelessness. One was a lack of ambition. A serene, self-contained, and frugal man, who grew up without material possessions and seems never to have acquired the taste for them, Gauss had little need of anyone’s approval and did not seek social advancement. The other factor, much

more common among mathematicians in all ages, was perfectionism. Gauss could not bring himself to present any result to the world until it was polished smooth, all in faultless logical order. His personal seal showed a tree with only sparse fruit, and the motto, Pauca sed matura—“Few, but ripe.”

This is, as I said, a common failing among mathematicians and often makes the reading of published mathematical papers a very tedious business. In one of the minor classics of modern psychological literature, The Presentation of Self in Everyday Life, Erving Goffman develops a theory of “performances,” in which a product or activity created in conditions of disorder and opportunity in some “back” environment is presented as a smooth, finished creation at the “front.” Restaurants illustrate the point. Dishes prepared in the clatter, breakage, and yelling of an overheated kitchen appear in the public area as flawless arrangements on spotless plates, delivered by dapper murmuring waiters. A great deal of intellectual work is like this. Says Goffman:

Published mathematical papers often have irritating assertions of the type: “It now follows that…,” or: “It is now obvious that…,” when it doesn’t follow, and isn’t obvious at all, unless you put in the six hours the author did to supply the missing steps and checking them. There is a story about the English mathematician G.H. Hardy, whom we shall meet later. In the middle of delivering a lecture, Hardy arrived at a point in his argument where he said, “It is now obvious that….” Here he stopped, fell silent, and stood motionless with fur-

rowed brow for a few seconds. Then he walked out of the lecture hall. Twenty minutes later he returned, smiling, and began, “Yes, it is obvious that….”

If he lacked ambition, however, Gauss also lacked tact. He made a great deal of trouble for himself with his fellow mathematicians by referring to discoveries he had made, but not published, years before someone else discovered and published them. This was not vanity— Gauss was free of vanity—but what Dr. Johnson called “stark insensibility.” In a book published in 1809, for example, Gauss referred to his discovery in 1794 of the method of least squares (a way of finding the best “fit” for a number of experimental observations). He had, of course, not published the discovery at the time he made it. The older French mathematician Adrien-Marie Legendre had discovered, and published, the method in 1806 and was furious at Gauss’s claim to prior discovery. There is no doubt of the truth of Gauss’s claim—we have documentary evidence—but if he wanted the credit, he really should have published. He did not care about the credit, though; and would not publish a paper if he hadn’t enough time to polish it to perfection.

III. In December 1849 Gauss exchanged letters with the astronomer Johann Franz Encke (after whom a famous comet is named). Encke had made some remarks about the frequency of primes. Gauss’s letter opened:

[My italics]

By “chiliads” Gauss meant blocks of 1,000 numbers. So beginning in 1792—when he was fifteen years old!—Gauss had amused himself by tallying all the primes in blocks of 1,000 numbers at a time, continuing up into the high hundreds of thousands (“without quite getting through a million”).

To get a feeling for the effort involved here, I set myself the task of extracting the primes from the chiliad 700,001 to 701,000, using just the aids that would have been available to Gauss: a pencil, some sheets of paper, and a list of the primes up to 829, which is as many as you need in order to apply the basic prime-finding process to numbers up to 701,000.¹⁷ I confess I gave up after an hour, when I had worked through prime divisors up to 47…which means I had 130 prime divisors still to go. You are welcome to try the same exercise yourself. This was Gauss’s “idle quarter of an hour” (unbeschäftigte Viertelstunde).

The sentence I italicized in the extract from Gauss’s letter to Encke is the first of the two PNT-related results I showed in Chapter 3.ix. It is, as I remarked there, equivalent to the PNT. There is no doubt that Gauss was indeed working on this in the early 1790s. His claim is well documented, just as other claims of the same kind were. He just never bothered to publish.

IV. Oddly, the first published work touching on the PNT came from that same Adrien-Marie Legendre who had been so vexed by Gauss’s claim to have discovered the method of least squares. In 1798—that is, five or six years after Gauss had unearthed the PNT, without making his results known to the world—Legendre published a book titled

Essay on the Theory of Numbers, in which he conjectured, on the basis of some prime counts of his own, that

for some numbers A and B, “to be determined.” In a later edition of the book he refined this conjecture (which he could not prove) to

where A, for large values of x, tended to some number near 1.08366. Gauss discusses Legendre’s conjectures in his 1849 letter to Encke. He demolishes the 1.08366 value but comes to no other very definite conclusions.

No doubt the Encke letter, if he had read it, would have caused poor Legendre to throw another conniption. Fortunately he had died some years before it was written.¹⁸

V. Because I am surveying here relevant discoveries and conjectures before 1800, and because he was the author of the “Golden Key,” of which I am going to make so much in later chapters, this is the right place to introduce the other first-rank mathematical genius born in the eighteenth century, Leonhard Euler (pronounced “oiler”). Euler (1707–1783) was, says E.T. Bell in Men of Mathematics, “probably the greatest man of science that Switzerland has produced” and he is, so far as I know, the only mathematician to have two numbers named after him: e, which I have already mentioned, equal to 2.71828…, and the Euler-Mascheroni number, which I have not had enough space to describe properly in this book,¹⁹ equal to 0.57721…. In order to introduce Euler, I must first open up a new geographical region in the history of this topic, Russia.

Russia, as I think is well known, entered the modern age somewhat behind the rest of Europe, and her entry was accomplished

mainly by the energy and imagination of Peter the Great, who as a boy of 10 was crowned Tsar in 1682. Peter’s regnal dates are commonly given as 1682–1725. In fact, for the first seven of those years, he reigned jointly with his blind, lame, and speech-impaired half-brother Ivan, and the government was actually controlled by Ivan’s sister, Sophia. Peter attained sole power only in 1689, at age 17. Even then he displayed no interest in statecraft and spent the next five years amusing himself. Fortunately he was a man of keen intelligence and great curiosity, and most of his amusements were of an improving sort. He was especially fond of the company of foreigners, of whom at that time there was a large settlement near Moscow, in the so-called “German suburb.” Here, among Scottish mercenaries, Dutch merchants, and German and Swiss engineers, Peter took in European science and culture and indulged his passion for fireworks and boats (in between riotous banquets and all-night drinking bouts). In 1692– 1693, at Lake Pleschev near Moscow, Peter actually built a warship himself, from the keel up. The following year, 1694, his mother died and Peter took power in earnest.

In 1695–1696 this extraordinary, extraordinary-appearing man—he stood 6 feet 7 inches and suffered from occasional, but terrifying, facial twitches—attacked the Black Sea port of Azov and wrested it from the Ottoman Turks. In 1697–1698 he traveled incognito in France, Britain, and Holland, the first Russian sovereign to go abroad, learning as he traveled. (From his British trip the following story is well known, though it is almost certainly apocryphal. Staying at John Evelyn’s country house outside London, Peter marched into the drawing-room one day with a shotgun over his arm and announced, in thick English, “I haff shot a peasant.” “No, no, my dear fellow,” replied his host, laughing. “You mean a pheasant.” “ Nyet,” said Peter, shaking his head. “It voss a peasant. He voss insolent, unt so I shot him.”) Returning to Russia, he began his great campaign of reform, ordering the nobility to shave their beards, humbling the Church and crushing the old Muscovite imperial guard, the Streltsy, which had terrorized his childhood. In 1700 he began his 20-year

war with Charles of Sweden; in 1703 he broke through Swedish territory and occupied the length of the river Neva, from Lake Ladoga to the shores of the Baltic. There, on land still in the legal possession of a powerful, undefeated enemy, on the boggy estuary of the Neva, he founded his new capital, St. Petersburg.

One of those astonishing personalities that put the lie to any notion of history as a mere mechanical shadow-play of impersonal forces, Peter went on to reform the government, the nobility, trade, education, and even the customary dress of his people. Not all of it worked; that is, not all of it “stuck”; and not all of it penetrated very far into the gloomy wooded depths of that vast old country; but there is no doubt that Peter left Russia a very different place from the one he found.

Most to the point so far as this book is concerned, he made her a nation hospitable to mathematics and mathematicians.²⁰

VI. In January 1724, Peter issued a decree establishing an Academy at St. Petersburg. The decree explained that in the normal way of things an academy, where learned scholars carried out research and produced inventions for the use of the state, was different from a university, which existed to teach young people. Because of the dearth of learning in Russia, however, the St. Petersburg Academy would include a university and a gymnasium (that is, a secondary school) under its authority. It would also have its own observatories, laboratories, workshops, publishing house, print shop, and library. Peter did not do things by halves.

The dearth of learning in Russia was indeed so great that there were no Russians capable of acting as academicians. In fact, since Russia lacked any significant number of elementary or secondary schools, there were not even any Russian youngsters qualified to attend as students at the attached university. These problems were solved by simply importing the required personnel. This was well-established prac-

tice in Europe. The first director of the Paris Academy of Sciences, founded 60 years earlier, had been the Dutch physicist Christiaan Huygens. St. Petersburg was a long way from the great European centers of culture, though, and Western Europeans still thought of Russia as a dark and barbarous land, so generous terms had to be offered. Eventually, though, it all got off the ground, the shortage of students for the university being solved by importing eight German youngsters. The St. Petersburg Academy opened its doors in August, 1725— too late for Tsar Peter to preside over the ceremony; he had died six months earlier.

Among the foreign scholars who showed up at the first session of the St. Petersburg Academy were two brothers, Nicholas and Daniel Bernoulli. Aged 30 and 25 respectively, they were sons of Johann Bernoulli of Basel in Switzerland—the gentleman we met in Chapter 1.iii in connection with the harmonic series. (There was a whole dynasty of mathematical Bernoullis; in this generation, in fact, there was a third brother, who followed his father into the chair of mathematics at Basel University, and who “personified the mathematical genius of his native city in the second half of the eighteenth century,” according to the Dictionary of Scientific Biography.)

Unfortunately, after less than a year in St. Petersburg, Nicholas Bernoulli died (“of a hectic fever”—D.S.B.), creating a vacancy at the Academy. Daniel Bernoulli had known Leonhard Euler in Basel and recommended him. Euler, glad of the chance of an academician’s post at such a young age, arrived in St. Petersburg on May 17, 1727, a month after his 20th birthday.

That date was also, unfortunately, 10 days after the death of Empress Catherine, Peter’s wife, who had succeeded Peter on the throne and followed through on his plans for the Academy. It was a bad time to come to Russia. The 15-year period between Peter’s death and the reign of Elizabeth, his daughter, was one of feeble leadership, clique politics, and occasional outbreaks of xenophobia. The warring cliques all maintained networks of spies and informers, and the atmosphere in the capital (which St. Petersburg now was) went from bad to worse.

Under the cruel and brutish Empress Anna, who reigned 1730–1740, Russia descended into one of those spells of state terrorism that she seems particularly prone to, with endless treason trials, mass executions, and other atrocities. This was the notorious Bironovschina, named for Anna’s favorite, the German Ernst Johann Biron, on whom ordinary Russians put the blame.

Euler stuck it out for 13 years, burying himself in work, staying well clear of the court and its intrigues. “Common prudence forced him into an unbreakable habit of industry,” writes E.T. Bell, and this seems as good an explanation as any for Euler’s astonishing productivity. Even now the full edition of his collected works is not complete. To date it comprises 29 volumes on mathematics, 31 on mechanics and astronomy, 13 on physics, and 8 volumes of correspondence.

For Euler’s friend Daniel Bernoulli, with whom he lodged during the early years in St. Petersburg, the stifling political atmosphere of Russia after Peter was all too much. In 1733 Daniel left to return to Basel, and Euler took over the chair of mathematics at the Academy. This brought him sufficient income to get married. He chose a Swiss girl, Catherine Gsell, whose father was a painter living in St. Petersburg.

It was in these circumstances that Euler solved the Basel problem in 1735; I’ll describe that problem in the next chapter. Two years later, in a small memorandum on infinite series, Euler discovered the result that I have called “the Golden Key,” and to which I have devoted the first half of Chapter 7. He was, in short, a principal player in the story I am telling—but this will emerge more clearly later, as the mathematical side of the story unfolds.

VII. By 1741 Euler had had enough of secret-police spies and the public impaling of “traitors.” Frederick the Great was now on the throne of Prussia and had already embarked on his plan to make the

Kingdom of Prussia—a mere duchy until 1700—one of the great powers of Europe. He planned an Academy of Science in Berlin to replace, or re-vivify, that city’s moribund Society of Sciences and invited Euler—by now famous throughout Europe—to be the Academy’s Director of Mathematics. Euler arrived in Berlin on July 25, 1741, after a one-month sea and land journey from St. Petersburg. Frederick’s mother, Sophia Dorothea of England—she was George II’s sister—took a shine to young Euler (he was still only 34) but could not get him to say much. “Why won’t you talk to me?” she asked him. Euler replied, “Because, Madame, I have come from a country where every person who speaks is hanged.”

In fact, part of Frederick’s aim in bringing Euler to Berlin was precisely that he should speak. Frederick wanted his court to be a sort of salon, full of brilliant people saying brilliant things to each other. Euler was a very brilliant man indeed, but unfortunately only in mathematics. His opinions on matters of philosophy, literature, religion, and worldly affairs, while well-informed and sensible, were commonplace and uninspired. Further, Frederick was a manipulative egotist who, while in principle wishing to surround himself with geniuses, in practice preferred second-raters who would flatter him. Setting aside a few luminaries like Voltaire and Euler, the general intellectual level at Frederick’s court was probably less than scintillating. In 1745–1747 Frederick built the Sans Souci summer palace for himself at Potsdam, 20 miles outside Berlin. (Euler helped design a system of water pumps for the place.) A visitor to Sans Souci asked one of the royal princes: “What do you do here?” The prince replied: “We conjugate the verb s’ennuyer.” S’ennuyer means “to be bored.” The language of Frederick’s court was French, the language of high society all over Europe.²¹

Euler stuck that out for 25 years, through all the horrors of the Seven Years War, when foreign armies twice occupied Berlin, and one in ten of Frederick’s subjects died of hunger, disease, or by the sword. By then a second Catherine, Catherine the Great, was on the throne of Russia. (It is interesting that for two-thirds of the eighteenth cen-

tury—67 years out of 100—Russia, one of the most difficult nations to govern, was ruled by women, for the most part very successfully.) Catherine showed every sign of being an enlightened monarch, firmly in control of her throne. She was, furthermore, a German princess, and it is possible Euler had some acquaintance with her at Frederick’s court before she was shipped off to St. Petersburg to marry Peter the Great’s grandson. Be that as it may, he left the genteel intrigues of Sans Souci to resume his position in St. Petersburg—which, incredibly, had been held open for him. He spent his last 17 years in Russia, productive to the end, and died in an instant, in full possession of all his powers but sight, at age 76, with a grandchild on his knee.

VIII. I have had to restrain myself considerably in this sketch of Leonhard Euler, because he is one of my favorite people in the history of mathematics for a number of reasons. One is that his work is a pleasure to read. Euler always expresses himself briefly and clearly, without any fuss, and without much of that polishing that Gauss went in for. Euler wrote mainly in Latin, but this is not much of an obstacle to appreciating him, as he had a spare and utilitarian style.²²

Euler’s crystal-clear Latin makes one realize what western civilization lost when scholars ceased writing in that language. Gauss was the last important mathematician to do so; this was one of those changes that came upon us after the Napoleonic wars. It is a curious thing that while the Congress of Vienna, which marked the end of those wars, was a gathering of reactionaries intent on restoring the status quo ante to Europe, in fact the wars had changed everything, and nothing could be the same after them. The historian Paul Johnson has written a good book about this, Birth of the Modern.

Another reason I find Euler so attractive is that, without being striking or eccentric or interesting in any particular way, he was a very admirable human being. When you read about his life you get a strong impression of serenity and inner strength. Euler lost the sight

in his right eye when he was barely 30 (the heartless Frederick called him “My Cyclops”) and went completely blind in his early 60s. Neither the partial nor the full disability seems to have slowed him down a bit. Of his thirteen children, only five survived into adolescence, and only three outlived him. His wife Catherine died when Euler was 69; a year later he remarried—to another Gsell, Catherine’s half-sister.

He loved children, and it is reported that he could do serious mathematics with infants playing at his feet. (As a writer working at home, with two small children running around, this is very impressive indeed to me.) He seems to have been incapable of intrigue, seems never to have lost a friend other than by death, and was frank in all his dealings—though, if Strachey is to be believed, willing to bend his principles a little for the sake of a quiet life.²³ He wrote one of the first pop-science bestsellers, Letters to a German Princess, explaining to ordinary readers why the sky is blue, why the moon looks larger when it rises, and similar points of common bafflement.²⁴

Underneath it all was a rock-solid religious faith. Euler had been raised a Calvinist and never wavered in his belief. His father, like Riemann’s, had been the pastor of a village church, and Euler, like Riemann, had originally been intended for a clerical career. We are told that while living in Berlin, “He assembled the whole of his family every evening, and read a chapter of the Bible, which he accompanied with an exhortation.” This, while attending a court at which, according to Macaulay, “the absurdity of all the religions known among men was the chief topic of conversation.” Hardworking, pious, stoical, devoted to his family, plain-living and plain-spoken—no wonder Frederick didn’t like him. But it is time to turn from the life to the work, and to look at Euler’s first great triumph, the Basel problem.