.A problem in number theory is as timeless as a true work of art. David Hilbert, Introduction to Legh Wilber Reid. The Elements of the Theory of Algebraic Numbers
Euclid in Alexandria. Euler in St Petersburg. The Göttingen trio – Gauss, Dirichlet, Riemann. The problem of prime numbers had been passed on like a baton from one generation to another. The new perspectives of each generation provided impetus for a fresh surge along the track. Each wave of mathematicians left its characteristic mark on the primes, a reflection of their era’s particular cultural outlook on the mathematical world. However, Riemann’s contribution took him so far ahead of the field that it would be nearly thirty years before anyone was in a position to capitalise on his rush of new ideas.
Then in 1885, out of nowhere, it looked as if the game was up. Although word got around more slowly than would Bombieri’s April Fool email over a century later, news began to spread that a little-known figure had not only picked up Riemann’s baton but had crossed the finishing line. A Dutch mathematician, Thomas Stieltjes, was boasting that he had a proof of Riemann’s Hypothesis – a proof confirming that all the zeros were on Riemann’s magic ley line, passing through .
Stieltjes was an unlikely victor. As a student he had failed his university examinations three times, much to the despair of his father, a member of the Dutch parliament and an eminent engineer responsible for building the docks of Rotterdam. But Thomas’s failure was not due to laziness. He had simply been distracted by the pleasure of reading real mathematics in Leiden’s library rather than mugging up on the technical exercises required for his examinations.
Gauss had been a favourite author of Stieltjes, and the Dutchman was eager to follow in the master’s footsteps. He took up a post at the observatory in Leiden, just as Gauss had worked at the observatory in Göttingen. The job had magically become available thanks to a word to the director of the observatory from Stieltjes’s influential father, but he was never aware of this helping hand. As he trained his telescope on the skies, it was the mathematics of celestial motion that captured his imagination rather than measuring the positions of new stars. As his ideas blossomed, he decided to write to one of the eminent mathematicians at the famous French academies, Charles Hermite.
Hermite was born in 1822, four years before Riemann. Now in his sixties, he had become one of the standard-bearers for Cauchy and Riemann’s work on functions of imaginary numbers. Cauchy’s influence on Hermite went beyond mathematics. As a young man Hermite had been an agnostic, but Cauchy, a devout Roman Catholic, had caught Hermite in a weak moment during a severe illness and had converted him to Catholicism. The result was a strange mix of mathematical mysticism, akin to the cult of the Pythagoreans. Hermite believed that mathematical existence was some supernatural state which mortal mathematicians were only occasionally allowed to glimpse.
Perhaps that’s why Hermite responded so enthusiastically to the letter sent by an obscure assistant at the Leiden Observatory, persuaded that this stargazer was blessed with a heightened mathematical vision. Soon they were conducting a furious mathematical correspondence which saw 432 letters exchanged over a period of twelve years. Hermite was impressed with the young Dutchman’s ideas, and despite Stieltjes being without a degree, gave him his support and ensured that Stieltjes was rewarded with a professorship at the University of Toulouse. In one letter to Stieltjes about his work, Hermite wrote, ‘Vous avez toujours raison et j’ai toujours tort.’ (‘You are always right and I’m always wrong.’)
It was during this correspondence that Stieltjes made his extraordinary claim to have proved the Riemann Hypothesis. Hermite’s faith in his young protégé gave him no cause to doubt that Stieltjes had indeed come up with a proof. After all, he had already made great contributions to other branches of mathematics.
Since Riemann’s conjecture had not had time to mature into the seemingly intractable challenge it currently represents, Stieltjes’s announcement was greeted with less excitement than it would have been today. Riemann had not trumpeted his hunch about the zeros, but had buried it deep in his ten-page paper with little evidence to support it. It would take a new generation to appreciate the importance of the Riemann Hypothesis. Stieltjes’s announcement was exciting nonetheless, because a proof of the Riemann Hypothesis would prove Gauss’s Prime Number Conjecture, which at the time was the Holy Grail of Number Theory. For numbers up to 1,000,000, Gauss’s guess at the number of primes was 0.17 per cent off the mark. By 1,000,000,000 the percentage error had fallen to 0.003 per cent. Gauss believed that, as the numbers got bigger and bigger, the percentage error would get smaller and smaller. By the late nineteenth century, Gauss’s conjecture had been around for long enough to earn its prospective conqueror great kudos. The evidence in support of Gauss’s hunch was certainly compelling.
By the time Stieltjes wrote to Hermite about his proof, the best progress towards cracking Gauss’s conjecture had been made in the 1850s in Euler’s old stomping ground of St Petersburg. The Russian mathematician Pafnuty Chebyshev couldn’t actually prove that the percentage error between Gauss’s guess and the true number of primes became smaller and smaller, but he did manage to show that the error for the number of primes up to N would never be more than 11 per cent, however big you chose N. This may sound a far cry from the 0.003% that Gauss had achieved for the number of primes up to a billion, but the significance of Chebyshev’s result was that he could guarantee that however far one counted primes, the error would not suddenly become overwhelmingly large. Before Chebyshev’s result, Gauss’s conjecture had been based solely on a small amount of experimental evidence. Chebyshev’s theoretical analysis provided the first real support for some connection between logarithms and primes. However, there was still a long way to go to prove that the connection would remain as tight as Gauss was conjecturing.
Chebyshev managed to achieve this control on the error by purely elementary means. Riemann, labouring away in Göttingen on his sophisticated imaginary landscape, knew of Chebyshev’s work. There is evidence of a letter that he had prepared to send to Chebyshev outlining his own progress; the surviving pages of Riemann’s notes include several drafts in which he tries out different spellings of the Russian’s name. It’s not clear whether Riemann did eventually send his letter to Chebyshev. With or without the letter, Chebyshev was never to improve his estimate of the error in counting the primes.
This is why Stieltjes’s announcement was still so exciting for mathematicians at the time. Nobody yet suspected how difficult the Riemann Hypothesis would be to prove, but proving Gauss’s conjecture was a recognised achievement. Hermite was keen to see the details of Stieltjes’s proof but the young man was rather reticent. The proof was not quite ready yet. Despite much prodding over the next five years, Stieltjes came up with nothing to support his claim. To try to counter his mounting frustration at Stieltjes’s reluctance to explain his ideas, Hermite came up with what he believed to be a cunning ruse to smoke out the proof. Hermite proposed that the Grand Prix des Sciences Mathématiques of the Paris Academy for 1890 be dedicated to a proof of Gauss’s Prime Number Conjecture. Hermite sat back, confident that the prize would be going to his friend Stieltjes.
This was Hermite’s plan: to win the prize it wasn’t necessary for Stieltjes to claim anything as impressive as having cracked the Riemann Hypothesis. Rather, he only needed to chart a small section of the imaginary landscape – the border between Euler’s landscape and Riemann’s extension. It was enough to prove there were no zeros on this line, running north—south through the number 1. Riemann’s landscape could be used to judge the errors in Gauss’s formula, which were determined by how far east each of Riemann’s zeros would be found in the landscape. The more easterly the zero, the bigger the error. The error would be very small if the Riemann Hypothesis were correct, but Gauss’s conjecture would still be true even if the Riemann Hypothesis were false – provided all the zeros lay strictly to the west of the north—south border through the number 1.
The deadline for the prize passed, and Stieltjes remained silent. But Hermite was not to be completely disappointed. Unexpectedly, his student Jacques Hadamard submitted an entry. Although Hadamard’s paper fell short of a complete proof, his ideas were enough to award him the prize. Spurred on by the award, by 1896 he had managed to fill in the gaps in his previous ideas. He couldn’t show that all the zeros were on Riemann’s critical line through , but he could prove that there were no zeros as far east as the border through 1.
Finally, a century after Gauss’s discovery of a connection between primes and the logarithm function, mathematics had a proof of Gauss’s Prime Number Conjecture. No longer a conjecture, thenceforth it was known as the Prime Number Theorem. The proof was the most significant result about the primes since the Greeks confirmed there were infinitely many such numbers. Even though we will never be able to count to the farthest reaches of the universe of numbers, Hadamard had proved that no surprises awaited any intrepid traveller. The early experimental evidence discovered by Gauss was not some misleading trick being played by Nature.
Hadamard could never have achieved what he did without Riemann’s head start. His ideas for the proof were steeped in Riemann’s analysis of the zeta landscape, yet he was still far from proving Riemann’s conjecture. In his paper explaining the proof, Hadamard acknowledges that his work didn’t match the achievement of Stieltjes, who until his death in 1894 was still claiming to have a proof of the Riemann Hypothesis. Stieltjes was the first in a long line of reputable mathematicians who have announced proofs but failed to deliver the goods.
Hadamard soon learnt that he would have to share the glory of proving the Prime Number Theorem. Simultaneously, a Belgian mathematician, Charles de la Vallée-Poussin, had also come up with a proof. Hadamard and de la Vallée-Poussin’s great achievement marked the start of a journey that would continue into the twentieth century, with mathematicians now eager to push on in their exploration of Riemann’s landscape. Hadamard and de la Vallée-Poussin had established the base camp in preparation for the major ascent towards Riemann’s critical line. It was during this period that the problem began to assume its position as the Everest of mathematical exploration even though, ironically, its proof depended on navigating the lowest points in the zeta landscape. With Gauss’s Prime Number Theorem finally claimed, it was time for Riemann’s great problem to emerge from the hidden depths of his dense Berlin paper.
It was another Göttingen resident, David Hilbert, who brought Riemann’s remarkable insight to the world’s attention. This charismatic mathematician helped more than anyone to launch the twentieth-century drive to claim the ultimate prize of the Riemann Hypothesis.
The town of Königsberg in Prussia had achieved some mathematical notoriety during the eighteenth century, thanks to the riddle about its bridges that Euler had solved in 1735. During the late nineteenth century the town made its way back onto the mathematical map as the birthplace of David Hilbert, one of the giants of twentieth-century mathematics.
Although Hilbert was very fond of his home town, he could see that it was within Göttingen’s city walls that the mathematical fire was burning brightest. Thanks to the legacy of Gauss, Dirichlet, Dedekind and, most of all, Riemann, Göttingen had become a mathematical Mecca. Perhaps more than anyone else at that time, Hilbert appreciated what a mathematical sea change Riemann had brought about. Riemann recognised that seeking to understand the structures and patterns underpinning the mathematical world was more fruitful than focusing on formulas and tedious calculations. Mathematicians began to listen to the mathematical orchestra in a new way. No longer obsessed with individual notes, they were starting to hear the underlying music of the objects they studied. Riemann had begun a renaissance in mathematical thinking that took hold in Hilbert’s generation. As Hilbert wrote in 1897, he wanted to implement ‘Riemann’s principle according to which proofs should be impelled by thought alone and not by computation’.
Hilbert made his mark in the academic circles of Germany by doing just that. He had learnt as a child that the Greeks had proved that there were infinitely many prime numbers that were required to build all possible numbers. He had read as a student that things appeared differently if you considered equations rather than numbers. It had become a challenge at the end of the nineteenth century to show that, in contrast to the primes, there were a finite number of equations that could be used to generate certain infinite sets of equations. Mathematicians of Hilbert’s day were trying to prove this by laboriously constructing the equations. Hilbert stunned his contemporaries with a proof that this finite set of building blocks must exist even though he couldn’t construct the set. Just as Gauss’s schoolteacher had looked on incredulously as the student slickly added the numbers from 1 to 100, Hilbert’s superiors seriously doubted that anything other than hard graft could explain the theory of equations.
It was quite a challenge to the mathematical orthodoxy of the day. If you couldn’t see the finite list, it was hard to accept its existence, even though the proof confirmed that it was there. To be told that something could not be seen but was unmistakably there was disconcerting for those still wedded to the French tradition of equations and explicit formulas. Paul Gordan, one of the experts in this field, declared of Hilbert’s work, ‘This is not mathematics. This is theology.’ Hilbert nonetheless stuck to his guns even though he was only in his twenties. It was eventually accepted that Hilbert was right, and even Gordan conceded the argument: ‘I have convinced myself that theology has its merits.’ Hilbert then turned to the study of numbers, something he described as ‘a building of rare beauty and harmony’.
In 1893 he was asked by the German Mathematical Society to write an account of the state of number theory at the close of the century. It was a daunting task for someone in his early thirties. A hundred years before, the subject had barely existed as a coherent entity. Gauss’s Disquisitiones Arithmeticae, published in 1801, had uncovered such fertile ground that by the end of the century number theory had blossomed so much that it had become overgrown. To help tame the subject, Hilbert was joined by an old friend, Hermann Minkowski. They had known each other ever since they were students together in Königsberg. Minkowski had made his mark in number theory by winning the Grand Prix des Sciences Mathématiques at the age of eighteen. He was only too happy to work on a project that would bring to life what he called ‘the insinuating melodies of this powerful music’. Their collaboration forged Hilbert’s passion for the primes, which Minkowski claimed would ‘wiggeln und waggeln’ under their spotlight.
Hilbert’s ‘theology’ earned him much respect amongst a number of influential mathematicians in Europe. In 1895 a letter arrived from Göttingen from one of the professors, Felix Klein, offering him a position in the hallowed university. Hilbert did not hesitate and accepted straight away. During the meeting to discuss his appointment, the faculty questioned Klein’s support and speculated that he was appointing a lackey who would not be able to hold his own. Klein assured them that, on the contrary, ‘I have asked the most difficult person of all.’ That autumn Hilbert made his way to the town where Riemann, his inspiration, had been a professor, hoping to further the mathematical revolution.
Before long the faculty became aware that Hilbert was not content simply to challenge mathematical orthodoxies. The professors’ wives were appalled by the behaviour of this newcomer to the faculty. As one of them wrote, ‘he is upsetting the whole situation here. I learned that the other night he was seen in some restaurants playing billiards in the backroom with the students.’ As time went by, Hilbert began to win the hearts of the Göttingen ladies, and got himself a reputation as a womaniser. At his fiftieth birthday party, his students performed a song with a verse for each letter of the alphabet detailing one of Hilbert’s conquests.
The bohemian professor acquired a bicycle to which he became deeply attached. He was often to be seen cycling through the streets of Göttingen carrying flowers from his garden for one of his flames. He would give lectures in his shirtsleeves, which was unheard of at the time. In draughty restaurants he would borrow feather boas from women diners. It wasn’t clear whether Hilbert was deliberately courting controversy or simply seeking the most obvious solution to every problem. Clearly, though, his mind was focused more on mathematical questions than on the niceties of social etiquette.
Hilbert set up a twenty-foot blackboard in his garden. On it, between tending his flower beds and performing stunts on his bicycle, he would chalk his mathematics. He loved parties and would always play his music loudly by choosing the largest available needle for his phonograph. When he eventually heard Caruso singing live he was quite disappointed: ‘Caruso sings on the small needle.’ But Hilbert’s mathematics far outstripped his personal eccentricities. In 1898, he shifted his attention away from number theory and turned instead to the challenge of geometry. He had become intrigued with new types of geometry proposed by several mathematicians during the nineteenth century which claimed to violate one of the fundamental axioms of geometry as proposed by the Greeks. Because of his intense belief in the abstract power of mathematics, the physical reality of objects was irrelevant to him, and he began to study the connections and abstract structures underlying these new geometries. It was the relationship between the objects that was important. Hilbert once famously declared that the theory of geometry would still make sense if points, lines and planes were replaced by tables, chairs and beer mugs.
A century before, Gauss had considered the challenge posed by these new models of geometry, but had held back from voicing such heretical thoughts. Surely it was impossible that the Greeks had got it wrong. Nevertheless, he had begun to question one of Euclid’s fundamental axioms of geometry, about the existence of parallel lines. Euclid had considered this question: if you draw a line and then a point off the line, how many lines are there parallel to the first line which run through the point? It seemed obvious to Euclid that the answer was that there was one and only one such parallel line.
At the age of sixteen, Gauss had already begun to speculate that there might be equally consistent and valid geometries in which there were no such things as parallel lines. In addition to Euclid’s geometry and this new geometry with no parallel lines, there might even be a third class of geometries in which there was more than one parallel line. If that were so, there would be geometries in which the angles of a triangle no longer added up to 180 degrees, which the Greeks had believed impossible. If there were several possible geometries, Gauss wondered, which of them best described the physical world? The Greeks had certainly believed that their model provided a mathematical description of physical reality. But Gauss was not at all convinced that the Greeks were right.
In later life, whilst surveying the state of Hanover, Gauss used some of the measurements he’d made in the neighbourhood of Göttingen to see whether a triangle of light beams shone between three hilltops might not contradict Euclid by having angles that did not sum to 180 degrees. Gauss thought that the line followed by a path of light might bend in space. Perhaps three-dimensional space was curved in the same manner as the two-dimensional surface of the globe. He had in mind the so-called great circles, such as lines of longitude, along which the shortest path between two points on the surface of the Earth is measured. In this two-dimensional geometry there are no parallel lines of longitude since they all meet at the poles. No one had contemplated the idea that three-dimensional space might also bend.
We realise now that Gauss was working on too small a scale to observe any significant bending of space to counter the view of a Euclidean world. Arthur Eddington’s confirmation of the bending of light from stars during the solar eclipse of 1919 supported Gauss’s hunch. Gauss never went public with his ideas, perhaps because his new geometries seemed to be at variance with the task of mathematics, which was to represent physical reality. The friends he did mention his idea to, Gauss pledged to secrecy.
The idea of these new geometries was eventually floated publicly in the 1830s by the Russian Nikolai Ivanovic Lobachevsky and the Hungarian János Bolyai. The discovery of these non-Euclidean geometries, as Gauss called them, did not rock the mathematical boat as much as Gauss had feared they might, but was simply dismissed as too abstract. As a result they were ignored for many years. Nonetheless, by Hilbert’s time they were beginning to emerge as a perfect expression of his own more abstract approach to the mathematical world.
Some mathematicians claimed that any geometry that did not satisfy Euclid’s assumption about parallel lines must contain some hidden contradiction that would cause it to collapse. As Hilbert began to explore this possibility, he saw that there was a strong logical bond between non-Euclidean and Euclidean geometry. He discovered the only way these non-Euclidean geometries could contain contradictions was if Euclid’s geometry also contained contradictions. This seemed like some kind of progress. Mathematicians at the time believed that Euclid’s geometry was logically sound. Hilbert’s discovery meant that these non-Euclidean models would have the same logical foundations. If one geometry collapsed, it would bring down all other geometries with it. But then Hilbert had a rather unsettling realisation. No one had actually proved that Euclid’s geometry had no hidden contradictions.
Hilbert began to think about how to go about proving that Euclid’s geometry did not contain contradictions. Although none had been found during the two thousand years since Euclid, that was not to say that they weren’t there. Hilbert decided that the first thing to do was to recast geometry in terms of formulas and equations. This practice had been initiated by Descartes (hence the name Cartesian geometry) and adopted by the French mathematicians of the eighteenth century. Geometry could be reduced to arithmetic via the use of equations which described lines and points, and a point could be changed into numbers describing its coordinates in space. Mathematicians believed that the theory of numbers contained no contradictions, so Hilbert hoped that by replacing geometry by numbers the question of whether Euclidean geometry contained contradictions could be settled.
However, instead of an answer to the problem, Hilbert found something even more unsettling: no one had actually proved that the theory of numbers itself did not contain contradictions. Suddenly Hilbert was reeling. The fact that mathematics had worked both theoretically and practically for centuries and had not produced contradictions had given mathematicians confidence in what they were doing. ‘Allez en avant, et la foi vous viendra’ (‘Go forward, and faith will come to you’) was the answer given by the eighteenth-century French mathematician Jean Le Rond d’Alembert to those who questioned the foundations of the subject. To mathematicians, the existence of the numbers they were studying was as real as the organisms that biologists were classifying. Mathematicians had been happily plying their trade, making deductions from the assumptions they believed were self-evident truths about numbers. No one had contemplated the possibility that these assumptions might lead to contradictions.
Hilbert had been pushed further and further back and was now having to question the very basis on which mathematics had been constructed. Now that the question had been asked, it was impossible to ignore these foundational problems. Hilbert himself believed that no contradictions would ever be discovered and that mathematicians were equipped to dispel any such doubts, to prove that the subject was built upon a solid bedrock. His question marked the coming of age of mathematics. The nineteenth century had seen the transition from mathematics as practical handmaiden to science to the theoretical pursuit of fundamental truths more akin to the philosophy of a former resident of Königsberg, Immanuel Kant. Hilbert’s deliberations on the very foundations of the subject gave him the platform from which to launch this new practice of abstract mathematics. His new approach would characterise mathematics during the twentieth century.
At the close of the year 1899, Hilbert was given the perfect opportunity to draw together the dramatic changes that his new ideas were bringing about in geometry, number theory and the logical foundations of mathematics. He received an invitation to deliver one of the main lectures to the International Congress of Mathematicians to be held in Paris the following year. It was a great honour for a mathematician who was still under forty.
Hilbert felt daunted by the task of addressing his community at the beginning of a new century. Surely it called for a truly momentous lecture, one that would live up to the occasion. Hilbert began consulting friends about his idea to use the lecture to speculate on the future of mathematics. This was highly unconventional and went against the unspoken rule that only complete, fully formed ideas were to be made public. It would take some nerve to forgo the security offered by presenting proofs of established theorems and, instead, to speculate on the uncertainties of the future. But Hilbert was never one to shy away from controversy. In the end he decided to challenge the international community with what hadn’t been proved rather than what had.
He still had his doubts. Was it wise to use the occasion to try something so new? Perhaps he should follow convention and talk about what he had achieved rather than what he couldn’t solve. Because of his procrastination he missed the deadline for submitting the title of his lecture and wasn’t listed as a speaker at the congress. By the summer of 1900 his friends were worried that he might miss out completely on this wonderful opportunity to present his ideas, but one day they all found on their desks the text of Hilbert’s lecture. It was entitled simply ‘Mathematical problems’.
Hilbert believed that problems are the lifeblood of mathematics but also that they should be chosen with care. ‘A mathematical problem should be difficult in order to entice us,’ he wrote, ‘yet not completely inaccessible, lest it mock at our efforts. It should be to us a guide post on the mazy paths to hidden truths, and ultimately a reminder of our pleasure in the successful solution.’ The twenty-three problems he had chosen to present were perfectly selected to meet his stringent criterion. In the sultry August heat in the Sorbonne in Paris, Hilbert rose to deliver his lecture and challenge the mathematical explorers of the new century.
In the late nineteenth century, many areas of study had been influenced by the distinguished physiologist Emil du Bois-Reymond’s philosophical movement which held that there are limits to our ability to understand nature. The catch phrase in philosophical circles had been ‘Ignoramus et ignorabimus’ – we are ignorant and we shall remain ignorant. But Hilbert’s dream for the new century was to sweep aside such pessimism. He ended his introduction to the twenty-three problems with a rousing battle cry: ‘This conviction of the solvability of every mathematical problem is a powerful incentive to the worker. We hear within us the perpetual call: There is the problem. Seek its solution. You can find it by pure reason, for in mathematics there is no ignorabimus.’
The problems which Hilbert set the mathematicians of the new century captured the revolutionary spirit of Bernhard Riemann. The first two problems on Hilbert’s list concerned the foundational questions that had begun to obsess him, but the others ranged far and wide over the mathematical landscape. Some of them were open-ended projects rather than questions that seemed as if they should have clear answers. They included one related to Riemann’s dream that the fundamental questions of physics would turn out to be answerable using mathematics alone.
The fifth problem arose out of Riemann’s notion that the different fields of mathematics, such as algebra, analysis and geometry, were intimately related and could not be understood in isolation from one another. Riemann had demonstrated how algebraic properties of equations could be deduced from the geometry of the graphs defined by those equations. It had taken some courage to oppose the dogma that algebra and analysis must be kept away from the potentially misleading power of geometry. This is why the likes of Euler and Cauchy were so against the graphical depiction of imaginary numbers. For them, imaginary numbers were solutions to equations such as x2 = −1 and should not be confused with pictures. But to Riemann it was obvious that the subjects were connected.
Hilbert mentioned Fermat’s Last Theorem in the build-up to the announcement of his twenty-three problems. Curiously, despite the public perception, even in Hilbert’s day, of this problem as one of the great unresolved questions in mathematics, it never featured as one of Hilbert’s choices. In Hilbert’s view it was ‘a striking example of the inspiring effect which such a very special and apparently unimportant problem may have on science’. Gauss had expressed the same sentiment when he declared that one could choose a host of other equations and ask whether these had solutions or not. There was nothing special about Fermat’s choice.
Hilbert took Gauss’s critique of Fermat’s Last Theorem as the inspiration for his tenth problem: is there an algorithm (a mathematical procedure that works something like computer software) that can decide in a finite amount of time whether any equation has solutions? Hilbert hoped that his question would move mathematicians’ attention away from the particular and persuade them to focus on the abstract. For example, Hilbert had always appreciated how Gauss and Riemann had inspired a new perspective on the primes. No longer were mathematicians obsessed with whether a particular number was prime – instead, they were seeking to understand the music that flowed through all the primes. Hilbert hoped that his question about equations might have a similar effect.
Although one reporter at the meeting described the resulting discussion as ‘desultory’, this was more to do with the oppressive August weather than the intellectual appeal of Hilbert’s lecture. As Hilbert’s closest friend Minkowski commented, ‘through this lecture, which indeed every mathematician in the world without exception will be sure to read, your attractiveness for young mathematicians will increase’. The risk Hilbert took in presenting such an unconventional lecture cemented his reputation in the twentieth century as a pioneer of new mathematical thinking. Minkowski believed that these twenty-three problems would prove to be hugely influential, and told Hilbert that ‘you really have taken a total lease on mathematics for the twentieth century’. His words turned out to be prophetic.
Within his list of broad open-ended problems there was one, the eighth, that was very specific: to prove the Riemann Hypothesis. In an interview, Hilbert explained that he believed the Riemann Hypothesis to be the most important problem ‘not only in mathematics but absolutely the most important’. In the same interview he was asked what he thought the most important technological achievement would be: ‘To catch a fly on the Moon. Because the auxiliary problems which would have to be solved for such a result to be achieved imply the solution of almost all the material difficulties of mankind.’ An insightful analysis, given the way the twentieth century panned out.
He believed that a proof of the Riemann Hypothesis would do for mathematics what Lunar fly-catching would do for technology. After posing the Hypothesis as his eighth problem, he went on to explain to the International Congress that a full understanding of Riemann’s prime number formula might put us in a position to understand many of the other mysteries of the primes. He mentioned both Goldbach’s Conjecture and the problem of the existence of infinitely many twin primes. The appeal of proving the Riemann Hypothesis was twofold: as well as closing a chapter in the history of mathematics, it would open many new doors.
Hilbert didn’t think the Riemann Hypothesis would remain unproved for so long. In a lecture he gave in 1919, he declared he was optimistic that he would live to see it cracked, and perhaps the youngest member of the audience would live to see Fermat’s Last Theorem proved. But he boldly predicted that no one then present would be alive to witness the solution of the seventh problem on his list – whether 2 to the power of the square root of 2 was the solution to an equation. Hilbert may have had great mathematical insight, but it wasn’t matched by his powers of prediction. Within ten years the seventh problem had fallen. It is also just possible that some young graduate at Hilbert’s 1919 lecture lived to witness Andrew Wiles’s proof of Fermat’s Last Theorem in 1994. Despite exciting progress over the last few decades, the Riemann Hypothesis might indeed still be unresolved when Hilbert wakes up, like Barbarossa, in five hundred years’ time.
There was one occasion when Hilbert thought that he wouldn’t have to wait so long. One day he received a paper from a student which purported to prove the Riemann Hypothesis. Before too long Hilbert found a gap in the proof, but the method impressed him. Tragically, the student died a year later and Hilbert was asked to give an address beside the grave. He praised the boy’s ideas and hoped that one day they might stimulate a proof of the great conjecture. Then with the words, ‘Consider if you will a function defined on imaginary numbers …’ in a wholly inappropriate departure that illustrates perfectly the stereotype of mathematicians disconnected from social reality, Hilbert launched into the details of the incorrect proof. Whether the story is true or not, it is believable. Mathematicians sometimes develop tunnel vision.
Hilbert’s lecture quickly pushed the Riemann Hypothesis into the limelight. It was now seen as one of mathematics’ greatest unsolved problems. Although his obsession with the Hypothesis produced no direct contribution to its solution, the new programme that Hilbert was proposing for twentieth-century mathematics would be deeply influential. Even his questions on physics and foundational questions about the axioms of mathematics would by the end of the century have played their part in improving our understanding of the primes. Meanwhile, though, Hilbert was responsible for bringing to Göttingen a mathematician who would be next in line to carry the baton passed from Gauss to Dirichlet to Riemann.
A position in Göttingen had become vacant following the tragically early death of Hilbert’s closest friend, Minkowski. Aged only forty-five, Minkowski suffered devastating appendicitis. Hilbert had just succeeded in solving Waring’s Problem, which had to do with writing numbers as sums of cubes, fourth powers and beyond. He knew that Minkowski would appreciate the result because it extended the work for which Minkowski had been awarded the Grand Prix des Sciences Mathématiques by the French Academy when he was only eighteen. ‘Even on the hospital bed, lying mortally afflicted, he was concerned with the fact that at the next meeting of the seminar, when I would talk on my solution of Waring’s Problem, he would not be able to be present.’
Minkowski’s death affected Hilbert deeply. As one student at Göttingen related, ‘I was in class when Hilbert told us about Minkowski’s death, and Hilbert wept. Because of the great position of a professor in those days and the distance between him and the students, it was almost more of a shock for us to see Hilbert weep than to hear that Minkowski was dead.’ Hilbert was keen to find a successor whose passion for number theory was the equal of Minkowski’s.
By all accounts, Hilbert’s choice, Edmund Landau, was not an easy man. It seems that it was a toss-up between appointing him or someone else. Hilbert asked his colleagues, ‘Who of the two is the most difficult?’ When the reply came that without doubt Landau was, Hilbert said that Göttingen must have Landau. Theirs was not to be a department of yes-men. Hilbert wanted colleagues who would challenge both social and mathematical conventions.
Landau was tough on his students and lived up to his billing as a prickly member of the department. His students used to dread weekend invitations to his house, where they had to humour his passion for mathematical games. A newly-wed student of Landau’s was just about to depart for his honeymoon. The train had almost left the station at Göttingen when Landau stormed down the platform, pushed the manuscript of his latest book through the carriage window and demanded, ‘I want it proof-read by the time you return!’
Landau soon assumed the mantle of successor to the tradition of Riemann and Gauss and was the central figure in Europe in developing the work of de la Vallée-Poussin and Hadamard. His temperament was perfectly suited to striking out from the base camp they had established and heading up the slopes of Mount Riemann. To prove Gauss’s Prime Number Theorem, Hadamard and de la Vallée-Poussin had shown that there were no zeros on the north—south border through the number 1. The challenge now was to prove there were no zeros before one reached Riemann’s critical line through .
Landau was joined in his expedition by Harald Bohr. Bohr was based in Copenhagen but was one of the regular pilgrims who made their way across Europe to Göttingen. Bohr’s brother, Niels, was ultimately to become world-famous as one of the creators of the theory of quantum physics. Harald had already made a name for himself as a key player in the Danish football team that secured silver at the 1908 Olympic Games.
Together, Landau and Bohr made the first successful push to navigate the points at sea level in Riemann’s landscape. They were able to show that most of the zeros like to be bunched up against Riemann’s ley line. They considered the number of zeros from 0.5 to 0.51 and compared it to the number of zeros outside this thin strip of land. They were able to prove that the zeros in this strip at least accounted for a large proportion of the zeros. Riemann had predicted that all the zeros were on the line through . Landau and Bohr couldn’t prove anything as definite as that, but they had made a start.
To make their argument work, the strip didn’t necessarily need to be of width 0.01. However narrow it was, even of width 1/1030, say, Landau and Bohr could still prove that most zeros were inside this vertical band of land. Yet, frustratingly, neither of them could show that this meant that most must actually be on Riemann’s line through , something which Riemann claimed he had proved but had never published. This may seem counter-intuitive. If all the zeros lie in a vanishingly small band, then why can’t we conclude that most of them must actually lie on the critical line? Such are the mysteries of mathematics. Suppose, for example, that for every number N, there are 10N zeros in the narrow band between 
and 
. Such a hypothetical set-up would satisfy the result established by Bohr and Landau without there requiring any of the zeros to lie on the critical line through .
Göttingen by this time was beginning to live up to the motto, blazoned across the town hall, declaring that outside its medieval walls there was no life. More than anything else, Hilbert’s influence turned Göttingen from Riemann’s quiet university town into the mathematical powerhouse it had become in the early twentieth century. In Riemann’s time it was Berlin that was buzzing with intellectual energy, but when Hilbert was later offered a position at Berlin University he turned it down. The small medieval town steeped in Gauss’s heritage was the perfect environment for mathematical activity.
Hilbert was able to bring the world’s best mathematicians to Göttingen thanks to money bestowed by a mathematics professor, Paul Wolfskehl, who died in 1908. In his will Wolfskehl had left 100,000 marks as a prize for the first person to come up with a proof of Fermat’s Last Theorem. This was the prize that Wiles had read about as a child and sparked his interest in trying to prove Fermat’s riddle. (The prize money that Wiles eventually received for his proof was significantly devalued by the hyperinflation in Germany that followed the two world wars.) Wolfskehl’s will stipulated that for every year that the theorem remained unproved, the interest accrued by the prize money should be used to fund visitors to Göttingen.
Landau took charge of checking the solutions sent to the faculty in Göttingen. Eventually the task became so overwhelming that Landau resorted to passing manuscripts to his students together with a standard letter of rejection for them to complete. It read, ‘Thank you for your solution of Fermat’s Last Theorem. The first mistake occurs on page … line …’ Hilbert assumed the much more pleasant job of spending the interest generated by the unclaimed prize money. It gave him the flexibility to bring many mathematicians to Göttingen, so much so that he hoped Fermat’s Last Theorem would remain unproved. ‘Why should I kill the goose that lays the golden eggs?’ he asked.
It was generally accepted that any young mathematician who wanted to make his way in the world first made his way to Göttingen. One student compared Hilbert’s influence on mathematics to listening to ‘the sweet flute of the Pied Piper … seducing so many rats to follow him into the deep river of mathematics’. Not unexpectedly, many of these mathematical rats came from the academies of Continental Europe that had blossomed in the political and intellectual revolutions that had swept through Europe in the nineteenth century.
In contrast, Great Britain was suffering from her traditional inability to absorb good ideas coming from the Continent. Just as the shores of England had remained remarkably resistant to the political turmoil of the French Revolution, mathematics in England had missed Riemann’s revolution. Imaginary numbers were still regarded as a dangerous Continental notion. Indeed, mathematics in England had not flourished significantly since the dispute in the seventeenth century between Newton and Leibniz over who should get the credit for discovering calculus. Even if Newton had been the first, his country’s mathematical development would for many years be hamstrung by its refusal to recognise the superiority of Leibniz’s development of the subject. However, things were about to change.
By 1914 Landau and Bohr had completed their work showing that most zeros were, at least, bunched up against Riemann’s critical line. But how far had mathematicians got in charting the zeros that were on the line? Out of the infinite number of points at sea level that mathematicians knew were there, only seventy-one had so far been identified as lining up along Riemann’s critical line.
Then came an important psychological breakthrough. After two centuries in the wilderness of disinterest in ideas from the Continent, an English mathematician, G. H. Hardy, seized Riemann’s baton and managed to prove that infinitely many of the zeros were indeed lining up on the north—south line running through . Hilbert was very impressed with Hardy’s contribution. Indeed, when Hilbert discovered that Hardy was having trouble with the authorities in Trinity College, Cambridge about his accommodation, he wrote a letter to the Master. Hardy, said Hilbert, was not only the best mathematician in Trinity, he was the best in England and he should therefore have the best rooms in the college.
Hardy’s fame outside mathematical circles owes much to his eloquent memoir A Mathematician’s Apology, but he earned his mathematical laurels for his contributions to the theory of prime numbers and the Riemann Hypothesis. If Hardy had proved that infinitely many zeros were on the line, then was the game up? Had Hardy proved the Riemann Hypothesis? After all, if there are infinitely many zeros and Hardy had proved that infinitely many of them were on Riemann’s line, aren’t we home and dry?
The infinite, unfortunately, is a slippery character. Hilbert liked to illustrate the mysteries of the infinite by using the idea of a hotel with an infinite number of rooms. You might check each odd-numbered room and find them all occupied, but even though you’ve checked an infinite number of rooms there are still all the even-numbered rooms to account for. In Hardy’s case, checking rooms to see whether they are occupied was replaced by checking zeros to see whether they were on the critical line. Unfortunately, Hardy had not even managed to prove that at least half the zeros were on the line. He’d accounted for an infinite number of rooms, but as a proportion of all the rooms left it represented zero per cent. Hardy’s achievement was extraordinary but there was still a long way to go. Hardy had taken a bite out of the zeros, but what he was left with was as enormous and intractable as before.
That tantalising first taste was to act like a drug for Hardy. Nothing would obsess him as much as his desire to prove that all the zeros were on Riemann’s line – with the possible exception of his passion for cricket and his running battle with God. As it was for Hilbert, the Riemann Hypothesis was at the top of Hardy’s wish list, as is clear from the New Year’s resolutions he wrote on one of the numerous postcards he sent to friends and colleagues:
(1) Prove the Riemann hypothesis.
(2) Make 211 [the first prime after the double century] 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 Mt. Everest.
(5) Be proclaimed the first president of the U.S.S.R., of Great Britain and Germany.
(6) Murder Mussolini.
Prime numbers had fascinated Hardy from an early age. As a child he amused himself in church by breaking down the numbers of the hymns into their prime building blocks. He loved to pore over books containing curiosities about these fundamental numbers, which he declared were ‘better than the football reports for light breakfast table reading’. In fact, Hardy believed that anyone who enjoyed reading the football would appreciate the joys of prime numbers. ‘It is a peculiarity of the theory of numbers that much of it could be published, and would win new readers for the Daily Mail.’ He believed that the primes retained enough mystery to intrigue the reader, yet were simple enough for anyone to begin to explore their magic. Hardy, more than any mathematician at the time, worked hard to communicate some of this passion for his subject, and did not believe it should remain the secret pleasure of those in the ivory towers of academe.
As the third of his New Year’s resolutions indicates, the church where he first cracked hymn numbers into primes also had a profound effect on Hardy. From an early age he became fiercely opposed to the idea of a God and the trappings of religion. He was to have a running battle with God through the whole of his life, attempting to prove his impossibility. His fight became so personal, he paradoxically conjured up the very character whose existence he vehemently wanted to deny. On trips to watch cricket he would take an anti-God battery to ward off any possibility of rain. Even though the sky was cloudless, he would arrive with four sweaters, an umbrella and a bundle of work under his arm. As he explained to his neighbouring spectators at the ground, he was trying to trick God into thinking that he hoped it was going to rain so that he could catch up on some work. God, his personal enemy, he believed would send sunshine to scupper any such plans Hardy had for doing mathematics.
One summer’s day, Hardy was frustrated to see the cricket match he was attending abruptly curtailed when the batsman complained of being unsighted by a flashing light emanating from the stands where he was sitting. His anger turned to joy when an enormous clergyman was asked to remove the huge silver cross from around his neck that was catching the light of the sun. Hardy could not contain himself and spent the lunch break firing off postcards to his friends describing cricket’s vanquishing of the clergy.
When the cricket season ended in September, Hardy would often visit Harald Bohr in Copenhagen before the English academic term began. They had a daily ritual of work. Every morning they would place a piece of paper on the table on which Hardy would write their task for the day: to prove the Riemann Hypothesis. Hardy had been hopeful that ideas developed by Bohr on his visits to Göttingen might provide a pathway to a proof. The rest of the day might be spent walking and talking, or scribbling away. Time after time their efforts failed to yield the breakthrough Hardy so hoped for.
Then, on one occasion, shortly after Hardy had set off on his return to England for the start of the new academic year, Bohr received a postcard. His heart raced as he read Hardy’s words: ‘Have proof of Riemann Hypothesis. Postcard too short for proof.’ Finally, Hardy had broken the impasse. The postcard, though, had a rather familiar ring to it. Fermat’s tantalising marginal comments flashed into Bohr’s mind. Hardy was too much of a prankster to have missed the irony in the postcard. Bohr decided to delay celebrations and await Hardy’s further elaboration. Sure enough, the postcard turned out not to be the breakthrough that Bohr had hoped for – Hardy was playing one of his games with God.
As Hardy boarded the ship to cross the North Sea from Denmark to England, the sea was unusually rough. The ship itself was not especially large, and Hardy began to fear for his life. So he took out his own very individual insurance policy. It was then that he sent Bohr that postcard announcing his fictional discovery. If the first passion in his life was to prove the Riemann Hypothesis, the second was his battle with God. Hardy knew that God would never allow the ship to sink and leave the world with the impression that Hardy and his proof had drowned and were lost for ever. His ploy worked, and he arrived safely back in England.
It is probably fair to say that Hardy’s addiction to the Riemann Hypothesis, combined with his colourful and charismatic character, helped to boost the problem to the top of mathematics’ most-wanted list. His eloquent writing style, encapsulated in A Mathematician’s Apology, was instrumental in promoting the importance of number theory and what he regarded as its central problem. It is striking that for all Hardy’s talk in the Apology of beauty and aesthetics in mathematics, the beauty of the proofs that Hardy was responsible for is often obscured by a mass of technical details that are required to see the proofs through to their conclusion. As often as not, success was a result not so much of a great idea as of hard graft.
The one book that was probably responsible for Hardy’s desire to become a mathematician was not a mathematics book at all. It was a story about the delights of a life at high table at Trinity College. The description of drinking port in the Senior Combination Room that he’d read in a novel, A Fellow of Trinity, fascinated him. Hardy admitted that he chose mathematics because ‘it is the one and only thing I can do at all well … until I obtained one, mathematics meant to me primarily a Fellowship at Trinity’.
To get there he was subjected to the gruelling round of examinations that the Cambridge system demanded. Hardy later realised that the emphasis the examination system placed on solving artificial technical problems and mathematical puzzles meant that, even after completing a degree in mathematics, few were aware what it was really all about. One of the Göttingen professors in 1904 parodied the problems that British students were expected to answer: ‘On an elastic bridge stands an elephant of negligible mass; on his trunk sits a mosquito of mass m. Calculate the vibrations of the bridge when the elephant moves the mosquito round by rotating its trunk.’ Students were expected to quote Newton’s Principia as if it were the Bible. Results were known by line number rather than by what they actually meant. Hardy believed that this system contributed to Britain’s time in the mathematical wilderness. British mathematicians were being taught to play their mathematical scales ever faster, but they were completely unaware of the beautiful mathematical music they could play once they had mastered their scales.
Hardy put his own mathematical enlightenment down to the French mathematician Camille Jordan’s book Cours d’Analyse, which opened his eyes to the mathematics that had been flourishing on the Continent. ‘I shall never forget the astonishment with which I read that remarkable work … and learnt for the first time as I read it what mathematics really meant.’
Hardy’s election to Trinity in 1900 released him from the burden of taking examinations, and set him free to explore the real world of mathematics.
Hardy was joined at Trinity in 1910 by a mathematician eight years his junior, J. E. Littlewood. Together they would spend the next thirty-seven years like a mathematical Scott and Oates, exploring the new land that had been opened up on the Continent. Their collaboration generated nearly a hundred joint papers. Bohr used to joke that there were three great English mathematicians during this period: Hardy, Littlewood and Hardy—Littlewood.
The two mathematicians each brought their own qualities to the collaboration. Littlewood was the bully boy who went in with all guns blazing in his assault on a problem. He revelled in the satisfaction of bringing a difficult problem to its knees. Hardy, in contrast, valued beauty and elegance. Invariably this carried over to the writing of their papers. Hardy would take Littlewood’s rough draft and would add what they called the ‘gas’ to produce the elegant prose that invariably accompanied their proofs.
It’s curious that these two mathematicians’ styles were mirrored in their physical appearance. Hardy was a beautiful man, one of those whose appearance maintains the stamp of youth well beyond their sell-by date. In his early days as a Fellow of Trinity he was often challenged by staff in the Senior Combination Room, who thought he must be an undergraduate who had lost his way in the labyrinth of Trinity’s corridors. Littlewood was rough-hewn – ‘a character straight out of Dickens’, as one mathematician remarked. He was strong and agile in mind and body. Like Hardy, he loved cricket and was a hard-hitting batsman. His other passion was music, something Hardy never felt an affinity for. As an adult he taught himself to play the piano, and had an intense love for the music of Bach, Beethoven and Mozart. He thought life too short to waste on lesser composers.
The other thing which separated them was sexuality. It was recognised that Hardy was very likely homosexual. Nevertheless, he was very discreet about it even though in Cambridge homosexuality was almost more acceptable than marriage. This was at a time when Oxford and Cambridge dons would have to leave their fellowships if they ever married. Littlewood declared Hardy to be a ‘non-practising homosexual’. By all accounts Littlewood was something of a ladies’ man. Although not up to Hilbert’s standards, he did become very friendly with a local doctor’s wife, with whom he spent summer holidays in Cornwall. Many years later, one of her children was looking into a mirror and commented on the striking resemblance to Uncle John. ‘That’s not surprising,’ she replied, ‘he’s your father.’
As befitting two mathematicians, Hardy and Littlewood’s collaboration was based on very clear axiomatic foundations:
Axiom 1: It didn’t matter whether what they wrote to each other was right or wrong.
Axiom 2: There was no obligation to reply, or even read, any letter one sent to the other.
Axiom 3: They should try not to think about the same things.
And the most important axiom of them all:
Axiom 4: To avoid any quarrels, all papers would be under their joint name regardless of whether one of them had contributed nothing to the work.
Bohr summed up their relationship thus: ‘Never was such an important and harmonious collaboration founded on such apparently negative axioms.’ Mathematicians today still talk about ‘playing under Hardy—Littlewood rules’ when they carry out joint work. Bohr found that Hardy remained true to his second axiom when they collaborated together in Copenhagen. He remembered the voluminous mathematical letters from Littlewood that arrived each day, and Hardy calmly tossing them into the corner of the room with a dismissive ‘I suppose I shall have to read them some day.’ When Hardy was in Copenhagen, there was only one thing on his mind: the Riemann Hypothesis. Unless Littlewood was sending him a proof of the Hypothesis, the letter flew into the corner.
There is a story recounted by Harold Davenport, a student of Littlewood’s, that Hardy and Littlewood almost fell out over the Riemann Hypothesis. Hardy had written a murder mystery in which one mathematician proves the Riemann Hypothesis, only to be killed by a second mathematician who then claims authorship of the proof. Littlewood was most upset. It wasn’t that Hardy had violated Axiom 4 and failed to include Littlewood as an author. Littlewood was convinced that the murderer was modelled on him, and he objected to the manuscript ever seeing the light of day. Hardy conceded, and mathematics was deprived of this literary gem.
Littlewood had come up through the ranks of Cambridge mathematical undergraduates, performing all the tricks required by the examination system. He made it to the top of the pile, earning himself the much coveted title of senior wrangler, jointly shared with another student, Mercer. The senior wranglers were celebrities in Cambridge, and photographs of them would be on sale at the end of the academic year. Perhaps his fellow students had already guessed that this was just the beginning of Littlewood’s outstanding career. When a friend tried to buy one of the photographs he was told, ‘I’m afraid we’re sold out of Mr Littlewood but we have plenty of Mr Mercer.’
Littlewood could see that the exams were not what mathematics was really about, but simply some technical game he was required to play and win before he could move on to the next stage. ‘The game we were playing came easily to me and I even felt some satisfaction in successful craftsmanship.’ Littlewood was eager to practise the craft he had learnt as an undergraduate and put it to more creative use. His introduction to serious mathematical research was to be something of a baptism of fire.
Fresh from the exams, Littlewood was keen to get stuck into research over the long summer vacation. He asked his tutor, Ernest Barnes, for a suitable problem on which he could cut his teeth. Barnes, who went on later to become bishop of Birmingham, thought for a while, and recalled an interesting function which no one had really got to grips with. Perhaps Littlewood could investigate where the function output zero. Barnes wrote out a definition of the function for Littlewood to take away for the summer. ‘It’s called the zeta function,’ said Barnes innocuously. Littlewood left Barnes’s rooms with the paper in his hand, oblivious of the fact that Barnes had just suggested he might like to spend the summer proving the Riemann Hypothesis.
Barnes had failed to provide Littlewood with the historical background of the problem, which would have indicated its difficulty. Littlewood’s tutor may even have been unaware that there were any connections between the zeros and prime numbers, and thought of it solely as an interesting problem: where does this function output the value zero? As Peter Sarnak, one of the leading lights in modern attempts on the Riemann Hypothesis, explains, ‘It was really the only analytic function mathematicians still did not understand as we entered the twentieth century.’ As Sir Peter Swinnerton-Dyer, who became one of Littlewood’s students, reflected at Littlewood’s memorial service, the fact that ‘Barnes thought [the Riemann Hypothesis] suitable for even the most brilliant research student, and that Littlewood should have tackled it without demur’ illustrated the dire state of British mathematics before Hardy and Littlewood had made an impact.
Littlewood battled all the summer, wrestling with the innocent-looking problem that Barnes had given him. Although he had no luck finding the locations of the zeros, he was very pleased with something else he came across. Just as Riemann had discovered some fifty years before, Littlewood realised that these zeros could tell you something about prime numbers. Although this had been known on the Continent since Riemann’s time, in England the connection between the zeta function and the primes was still unappreciated. Littlewood was thrilled by what he thought was a new link and in September 1907 he wrote it up for his dissertation in support of a research fellowship at Trinity. That Littlewood thought his discovery was original is further confirmation of how isolated British mathematics had become.
Hardy, who was one of the few in England aware of the recent progress made by Hadamard and de la Vallée-Poussin, knew that the result was not as original as Littlewood had hoped. Nonetheless, Hardy recognised Littlewood’s potential, and although he failed that year to be elected a Fellow of the college, there was a gentleman’s agreement to elect him next time round. He joined Hardy at Trinity in October 1910.
Cambridge was beginning to blossom as it opened its doors to the influences of the intellectual tradition across the Channel. Travel between the Continent and England was becoming easier, and Hardy and other academics were making the effort to visit many of the European centres of learning. The new contacts they made encouraged a flow of new journals, books and ideas from abroad. Trinity College in particular became an extraordinarily vibrant community in the early twentieth century. The Senior Combination Room was no longer a gentleman’s club, but a place of research. Conversation at high table did not confine itself to port and claret but was infused with the ideas of the day. Also at Trinity, working alongside Hardy and Littlewood, were the two most eminent philosophers active in England: Bertrand Russell and Ludwig Wittgenstein. Both were wrestling with the same foundational problems of mathematics that had so concerned Hilbert. And Cambridge was buzzing with new breakthroughs in physics made by the likes of J. J. Thomson, who was awarded a Nobel prize for his discovery of the electron, and Arthur Eddington, who had confirmed Gauss and Einstein’s belief that space was indeed curved and non-Euclidean.
The great collaboration between Hardy and Littlewood was fuelled by the timely arrival from Göttingen of a book by Landau about prime numbers. The publication in 1909 of his two-volume work Handbuch der Lehre von der Verteilung der Primzahlen (‘Handbook of the Theory of the Distribution of Prime Numbers’) proselytised the wonders of the connections between primes and the Riemann zeta function. Before Landau’s book, the story of Riemann and the primes was largely unknown in the broader mathematical community. As Hardy acknowledged in his obituary of Landau (jointly written with Hans Heilbronn), ‘The book transformed the subject, hitherto the hunting ground of a few adventurous heroes, into one of the most fruitful fields in the last thirty years.’ It would be Landau’s book that would in 1914 inspire Hardy to prove that infinitely many zeros sat on Riemann’s critical line. Fired up by the experiences of wrestling with the zeta function as a student, Littlewood too was encouraged to make the first of his great contributions to the subject.
To prove a theorem that Gauss believed to be true but couldn’t prove is generally regarded as a true test of a mathematician’s mettle. To disprove such a theorem puts one in a different league altogether. It is not often that Gauss had a hunch which turned out to be false. He had produced a function, the logarithmic integral Li(N), that he had predicted would guess the number of primes up to any number N, with increasing accuracy as N got bigger. Hadamard and de la Vallée-Poussin had carved their names into mathematical history by proving Gauss right. But Gauss had made a second conjecture: that his guess would always overestimate the number of primes – it would never predict that there were fewer primes than there really were in the range from 1 to N. This contrasted with Riemann’s refinement, which fluctuated between overestimating and underestimating the correct number of primes.
By the time Littlewood started to think about Gauss’s second conjecture, it had been confirmed to be true for all numbers up to 10,000,000. Any experimental scientist would have accepted 10 million pieces of evidence as utterly convincing support for Gauss’s hunch. Sciences with less of an addiction to proof and more respect for experimental results would have been perfectly happy to accept Gauss’s conjecture as a foundation stone upon which they could start to build new theories. By Littlewood’s day, some hundred years later, the mathematical edifice might well have towered way above this foundation. But in 1912 Littlewood discovered that, contrary to expectations, Gauss’s hypothesis was a mirage. The foundation stone crumbled into dust under his scrutiny. He proved that as you counted higher you would eventually come to regions of numbers where Gauss’s guess would switch from overestimating to underestimating the number of primes.
Littlewood also succeeded in demolishing another idea that was beginning to get a toehold. Many believed that Riemann’s refinement of Gauss’s guess at the number of primes would always be the more accurate. Littlewood showed that Riemann’s refinement might look more accurate as we count through the first million numbers, but in the farther reaches of the universe of numbers Gauss’s guess would sometimes give the better prediction.
Littlewood’s discoveries were particularly striking because Gauss’s guess starts to underestimate the number of primes only in regions of numbers that we will probably never be able to calculate. Littlewood could not even predict how far we would need to go before we could observe any of these phenomena. Indeed, to this day no one has actually counted far enough to arrive at a region of numbers where Gauss’s guess underestimates the primes. It is only through Littlewood’s theoretical analysis and the power of mathematical proof that we can be sure that somewhere along the line Gauss’s original prediction is false.
Some years later, in 1933, a graduate student of Littlewood’s named Stanley Skewes estimated that by the time one had counted the primes up to 
, one will have witnessed Gauss’s guess finally underestimate the number of primes. That is a ridiculously large number. Encounters with large numbers often elicit comparisons with the number of atoms in the visible universe, which is according to the best estimates approximately 1078, but the number suggested by Skewes defies even that. It is a number that begins with a 1 and then has so many zeros after it that even if you wrote a 0 on each atom in the universe you still wouldn’t have got anywhere near it. Hardy was to declare that the Skewes Number, as it became known, was surely the largest number that had ever been contemplated in a mathematical proof.
The proof of Skewes’s estimate was interesting for another reason. It is one of the many thousands of proofs which begin ‘suppose the Riemann Hypothesis is true’. Skewes could make his proof work only by assuming that Riemann’s conjecture is correct: that all the points at sea level in the zeta landscape are on the line through . Without making this assumption, mathematicians in the 1930s were unable to guarantee how far we would need to count before Gauss’s guess underestimated the number of primes. In this particular case mathematicians finally found a way to avoid having to cross the summit of Mount Riemann. In 1955 Skewes produced an even larger number that would still work in the event that the Riemann Hypothesis turned out to be false.
It was curious that, in contrast to their reluctance to accept Gauss’s second conjecture, mathematicians were beginning to have sufficient faith in the truth of the Riemann Hypothesis that they were prepared to build upon it while it remained unproved. The Riemann Hypothesis was now becoming an essential structural component in the mathematical edifice. But it was probably as much a matter of pragmatism as of faith. More and more mathematicians were finding themselves coming up against the Riemann Hypothesis as an obstacle to their mathematical progress. Only by assuming that it was true could they proceed any further. But as Littlewood illustrated with Gauss’s second conjecture, mathematicians have to be prepared for the possible collapse of all that is built on the foundations of the Riemann Hypothesis, should someone discover a zero off the line.
Littlewood’s proof had a huge psychological effect on the perception of mathematics and especially on the appreciation of the primes. It sent out a stark warning to anyone impressed by a vast accumulation of numerical evidence. It revealed that prime numbers are masters of disguise. They hide their true colours in the deep recesses of the universe of numbers, so deep that witnessing their true nature may be beyond the computational powers of humankind. Their true behaviour can be seen only through the penetrating eyes of abstract mathematical proof.
Littlewood’s proof also provided the perfect ammunition for those who argued that mathematics differs in some essential way from the other sciences. No longer could mathematicians be happy with the experimentalism of the seventeenth- and eighteenth-century brand of mathematics in which theories were advanced after minimal calculations. Empiricism was no longer a suitable vehicle in which to navigate the mathematical world. Millions of pieces of data might be sufficient evidence on which to base theories in the other sciences, but Littlewood had proved that in mathematics that would be treading on thin ice. From now on, proof was everything. Nothing could be trusted without conclusive evidence.
As more mathematicians found themselves forced to assume the truth of the Riemann Hypothesis, it became more imperative than ever to make sure that in some distant part of Riemann’s landscape there weren’t zeros straying off the critical line. Until that had been done, mathematicians would always live in fear that the Riemann Hypothesis might be disproved.