Since the mathematical sciences are so vast and varied, it is necessary to localise their cultivation, for all human activity is tied to places and persons. David Hilbert, speaking at a party to celebrate Landau’s arrival in Göttingen as a professor in 1913
Landau’s father, Leopold, discovered that there was a young mathematical prodigy living on his street in Berlin. He was intrigued and sent him an invitation to come to his house for tea. Although Carl Ludwig Siegel was rather shy, he agreed to go and meet the great Göttingen mathematician’s father. In his library, Landau senior took down the two volumes on prime numbers written by his son and handed them to Siegel. It would probably be too difficult for Siegel now, he explained, but perhaps later he might be able to read it. Siegel was to treasure Edmund Landau’s book, which would have a lasting impact on his mathematical development.
Siegel’s coming of age coincided with the start of the First World War. The idea of serving in the army filled the young, reticent boy with dread. He began to develop a deep-felt loathing of all things military. Despite the interest that Landau’s father had taken in his mathematical development, Siegel had initially opted to study astronomy in Berlin, believing that the subject couldn’t possibly have any relevance to war. But the astronomy course started late, so to fill the time he attended some mathematics lectures. It wasn’t long before he’d caught the bug. Exploring the universe of numbers became Siegel’s passion. Soon he was equipped to make sense of the volumes on prime numbers that Landau’s father had given him.
By 1917, the war inevitably encroached on Siegel’s life, and when he refused to serve in the army he was confined to a mental institution as punishment. Landau’s father intervened to get him released. ‘If it had not been for Landau I would have died,’ Siegel later admitted. In 1919 the young man, still recovering from the ordeal, joined his mathematical idol Landau at Göttingen, where his mathematical talents were to flower.
Siegel found he had to put up with Landau’s rather infuriating personality. On one occasion, when Siegel was already a senior mathematician, he visited Landau in Berlin. During dinner the professor spent the whole meal painstakingly explaining an extremely detailed and technical proof, insisting on providing the minutest detail. Siegel listened patiently, but by the time Landau had finished it was so late that he had missed the last bus home and was forced to walk all the way back to his lodgings. During the long walk he thought about Landau’s proof, which concerned points at sea level in a landscape similar to the one constructed by Riemann. By the time he had arrived home he had come up with a slick alternative proof to the laborious one that had made him miss his bus. The next day, in a moment of chutzpah, Siegel sent Landau a postcard thanking him for dinner and giving the succinct details of his alternative proof – all of which fitted on the same card.
When Siegel arrived in Göttingen, Germany was weighed down by the costs of war reparations, and he was forced to lodge with one of the professors in the department. Another professor bought him a bicycle so that he could pedal round the town’s medieval streets. At first Siegel was rather intimidated by the Göttingen mathematical hierarchy, especially the great Hilbert. So he worked quietly alone, determined to make the breakthrough that would impress the great names he passed in the corridors of the department. He sat in Hilbert’s lectures, soaking up the formidable man’s ideas. He knew that answering just one of Hilbert’s twenty-three problems would be his passport to success.
At first he was far too timid in the presence of giants such as Hilbert to air his ideas. He eventually plucked up the courage when several members of the senior faculty invited him to join them on a swimming trip to the River Leine. Hilbert appeared far less intimidating in a swimming costume, and Siegel felt brave enough to share with Hilbert his thoughts on the Riemann Hypothesis. Hilbert responded very enthusiastically, and his support for the shy mathematician secured Siegel a position at the University of Frankfurt in 1922.
During his lifetime, Siegel successfully contributed to a number of Hilbert’s problems, but it was his unconventional breakthrough on the eighth problem, the Riemann Hypothesis, that stamped his name firmly on the mathematical map.
Rethinking Riemann
As Siegel began to apply himself to the solution of Hilbert’s eighth problem, he was becoming aware that some mathematicians were growing disillusioned about Riemann’s contribution to the subject. Siegel’s mentor, Landau, was perhaps the most vocal critic of what Riemann had actually managed to achieve in his ten-page paper published in 1859. Although he acknowledged it to be a ‘most brilliant and fruitful paper’, he went on to qualify his praise: ‘Riemann’s formula is far from the most important thing in prime number theory. He just created the tools which when refined made it possible later to prove many other things.’
Meanwhile, in Cambridge, Hardy and Littlewood were becoming equally dismissive. By the late 1920s, Hardy’s inability to solve the Riemann Hypothesis was beginning to frustrate him. Littlewood too began to wonder whether the fact that they couldn’t prove it meant that it wasn’t actually true:
I believe this to be false. There is no evidence whatever for it. One should not believe things for which there is no evidence. I should also record my feeling that there is no imaginable reason why it should be true … Nonetheless life would be more comfortable if one could believe firmly that the hypothesis is false.
Riemann had indeed been rather lacking when it came to providing evidence that the zeros were located where his Hypothesis predicted. In his ten-page paper there wasn’t a single calculation of one of these points at sea level. Hardy believed that Riemann’s hunch about the zeros in his landscape was nothing more than heuristic speculation.
The fact that in his paper Riemann appeared not to have calculated the locations of these zeros contributed to the image of him as the thinking man’s mathematician, a man of ideas not interested in getting his hands dirty doing calculations. After all, this was the ethos of the revolution Riemann had initiated. Hilbert had similarly dedicated his life to promoting this new approach to mathematics. As he wrote in one of his papers, ‘I have tried to avoid the huge computational apparatus of Kummer [Ernst Kummer, Dirichlet’s successor in Berlin], so that here too Riemann’s principle should be realised, according to which proofs should be impelled by thought alone and not by computation.’ Hilbert’s colleague in Göttingen, Felix Klein, was fond of saying that Riemann worked primarily by means of ‘great general ideas’ and ‘often relied on his intuition’.
Hardy, however, was not content to rely on intuition. He and Littlewood managed to develop a method for calculating precisely the locations of some of the early zeros. If the Riemann Hypothesis was false, then, armed with their formula, there was a very small chance that they might quickly locate a zero that was not on Riemann’s critical line. The method they developed exploited the symmetry that Riemann had discovered in his landscape between the land to the east and the land to the west of the ley line passing through 
. They used their method in conjunction with an efficient means devised by Euler to approximate the value of infinite sums of numbers. By the end of the 1920s the Cambridge mathematicians had successfully located 138 zeros. As Riemann had predicted, they were indeed all on the line through . It was clear, however, that Hardy and Littlewood’s formula was running out of steam. It was becoming computationally unfeasible to pinpoint exactly the location of any of the zeros to the north of these 138.
It seemed that these calculations could not be pushed much further. Hardy had proved by theoretical analysis infinitely many zeros had to be on the line. Now there was a growing feeling that any zero that might be off the line wouldn’t be seen until one had travelled very far north in the landscape. As Littlewood had demonstrated, prime numbers, more than any other creature in the mathematical zoo, like to hide their true colours in the far distant reaches of the universe of numbers. As a result, mathematicians began to give up on explicitly locating zeros and began to concentrate on other, more theoretical features of the landscape that might reveal the mysteries of Riemann’s thinking.
All this was changed by a most unexpected discovery. While Siegel was struggling in Frankfurt with his ideas on the Riemann Hypothesis, he received a letter from the mathematical historian Erich Bessel-Hagen who had been working through Riemann’s unpublished notes. Riemann’s wife, Elise, had rescued some of them from the zealous housekeeper who had burnt many of his papers. She had given the majority of Riemann’s scientific papers to Riemann’s contemporary Richard Dedekind, but a few years later she began to regret having handed over anything that might contain personal details. She asked Dedekind to return them. Even if a manuscript was mostly mathematics, if it contained the slightest hint of a shopping list or the name of a family friend, Elise wanted the offending pages returned.
The remaining scientific papers were eventually deposited by Dedekind in the library in Göttingen. Bessel-Hagen had been trying to make sense of the mass of papers contained in the archives, but with little success. As with most mathematicians’ private jottings, they were a chaotic jumble of half-formed ideas and formulas. Bessel-Hagen wondered whether Siegel might do better at decoding these hieroglyphics.
Siegel wrote to the librarian in Göttingen to ask whether he could consult Riemann’s Nachlass, as it is now known. The librarian arranged for the documents to be sent to Siegel’s local library in Frankfurt. Siegel was looking forward to the task: it would be a welcome distraction from the frustrations of the lack of progress he was making in his research. The package duly arrived, and he hurried down to the library with a visiting colleague. As he opened the package, out spilled a mass of papers crammed with complicated numerical calculations. These pages would give the lie to the picture of Riemann as a man who for seventy years had been billed as a mathematician of intuition and concepts unable to provide much hard evidence to back his ideas. Pointing to the mass of calculations, Siegel exclaimed ironically, ‘Here are Riemann’s great general thoughts!’
Several minor mathematicians had previously rummaged through these pages in search of a clue to Riemann’s Hypothesis, but none could make sense of the mass of fragmented equations. Most baffling was the huge amount of sheer arithmetic computation that Riemann seemed to have done in his spare time. What were all these sums? It took a mathematician of the stature of Siegel to see what Riemann had been doing.
As Siegel gazed at these pages, he began to see that Riemann had held true to his teacher’s dictum. As Gauss had always stressed, an architect removes the scaffolding once he has erected the building. The brittle leaves that Siegel now held in his hands were packed to the edges with calculations. Riemann had been poor, having had to support his sisters in later life, and he could only afford low-quality paper, from which he squeezed every last space. Hilbert’s thinker turned out to be a master-calculator, and it was upon those calculations that he had built his conceptual view of the world, finding patterns in the evidence he collected. Some of the calculations were not innovative, for example the square root of 2 calculated to 38 decimal places, but Siegel was intrigued by others the like of which he had not seen anywhere before. As he scoured the pages, the chaotic jumble of random calculations began to make some sense. He realised that Riemann was calculating zeros.
Siegel discovered that Riemann was using an extraordinary formula which enabled him to calculate the heights in his zeta landscape very accurately. The first part of the formula was based on a trick that Hardy and Littlewood had discovered. Riemann had anticipated their contribution by some sixty years. The second piece of the formula was completely new: Riemann had also discovered a way to add up the remaining infinite sum that was much cleverer than the method currently being used. In contrast to Euler’s methods that had been used to locate the first 138 zeros, the points at sea level in the zeta landscape, Riemann’s formula would maintain a head of steam as he calculated farther north.
Sixty-five years after Riemann’s death, the august mathematician was still streets ahead of the competition. Hardy and Landau had been wrong to believe that Riemann’s paper was just a remarkable set of heuristic insights. Instead it was based on solid calculation and theoretical ideas that Riemann had chosen not to reveal to the world. Within a few years of Siegel’s discovery of Riemann’s secret formula, it would be used by Hardy’s students in Cambridge to confirm that the first 1,041 zeros were on Riemann’s line. The formula, however, would truly come into its own with the dawn of the computer age.
It is rather odd that it took mathematicians so long to realise that Riemann’s notes might contain such gems. There are certainly clues in Riemann’s ten-page paper, and in letters he wrote to other mathematicians at the time, that he was sitting on something. In the paper he mentions a new formula but goes on to say that he ‘has not yet sufficiently simplified it to announce it’. The mathematicians in Göttingen had been poring over this published paper for seventy years and, unbeknownst to them, a few blocks down the road was the magic formula for locating zeros. Klein, Hilbert and Landau were all happy to pass judgement on Riemann, though none of them had so much as glanced at the unpublished Nachlass.
In fairness, a glance at Riemann’s jottings is enough to indicate the magnitude of the task. As Siegel wrote, ‘No part of Riemann’s writings related to the zeta function is ready for publication; occasionally one finds disconnected formulas on the same page; frequently just one side of an equation had been written down.’ It was like poring over the first draft of an unfinished symphony. The final composition owes much to Siegel’s mathematical virtuosity in extracting the formula from the mess of Riemann’s notes. It fully deserves the name by which it is now known – the Riemann—Siegel formula.
Thanks to Siegel’s perseverance, a new side to Riemann’s character had been revealed. Riemann had certainly championed the importance of abstract thinking and general concepts. But he knew it was important not to neglect the power of computation and numerical experiment. Riemann had never forgotten the eighteenth-century tradition from which his mathematics had emerged.
The Nachlass housed in the library in Göttingen was only part of what was rescued from Riemann’s housekeeper. Elise Riemann wrote to Dedekind on May 1, 1875, describing some of the personal material she wanted back in family hands. This included ‘a small black book containing records of Riemann’s sojourn in Paris in spring 1860’. Only a few months before, Riemann had published his great ten-page paper on the primes, rushing to get it into print so that it would coincide with his election to the Berlin Academy. Now in Paris, after the flurry of publication, he had time to flesh out his ideas. The weather there was appalling; hail and snow prevented Riemann from exploring the city. Instead, he sat in his room, setting down his thoughts on paper. It is not unreasonable to speculate that, along with his personal impressions of Paris, in that little ‘black book’ Riemann recorded his thoughts about the points at sea level in his zeta landscape. The book has never been recovered, although there are a few clues to its fate.
Riemann’s son-in-law wrote to Heinrich Weber on July 22, 1892, that ‘at first mother could not come to terms with the idea that Riemann’s papers should no longer remain in private hands; to her, they are something sacred and she doesn’t like to think of them being made accessible to any student, who would then also be able to read the marginal notes, some of which are purely personal’. Unlike Fermat, whose nephew was only too keen to publish his uncle’s marginalia, Riemann’s family was reluctant to make public notes that Riemann never intended to publish. It seems then that at this stage the little black book was still in family hands.
Speculation about the location of this notebook is rife. There is evidence that Bessel-Hagen subsequently acquired some of the remaining unpublished material that had remained in family hands. It is unclear whether he bought it at auction or acquired it through a personal contact. Some of the papers found their way into the archive at the University of Berlin, but it appears that Bessel-Hagen decided to hang on to the rest of his collection. He died of starvation in the winter of 1946, in the chaos that followed the end of the Second World War. His belongings have never been found.
Another story has the little black book finding its way into Landau’s hands. It is said that, in view of the uncertainty of the inter-war years, he gave the notebook to his mathematical son-in-law, I.J. Schoenberg, who escaped to America in 1930. Again the trail runs dry. Given that there is now a million dollars at stake, the search for Riemann’s little black book has turned into a treasure hunt.
Without Riemann’s crib-sheet and Siegel’s determination, how long might it have taken us to unearth the magic formula? It is so sophisticated that we might still not have known it today. What other gems have we been deprived of because of the disappearance of that black book? Riemann claimed he could prove that most zeros were on the critical line, yet no one has yet matched that claim with a proof. What might there be still buried in the archives of German libraries? Did the black book find its way to America? Did it survive the housekeeper’s fire only to be lost in the fires of the Second World War?
By 1933, mathematicians across Germany were finding it increasingly difficult to concentrate on mathematics. The swastika was flying over the library in Göttingen. The faculty was home to many Jewish and left-wing mathematicians. Street campaigns at the time specifically focused on the mathematics department as a ‘fortress of Marxism’ and by the mid-thirties most of the faculty lost their jobs in Hitler’s purge of the universities. Many of them sought refuge overseas. Landau, although he was Jewish, was allowed to stay because he had been appointed before the outbreak of the First World War. The non-Aryan clause in the civil-service law of April 1933 did not apply to long-serving professors or those who had fought in the war.
Things got worse. By the winter of 1933, Landau’s lectures were being picketed by Nazi students, including one of the most brilliant mathematicians of his generation, Oswald Teichmüller. A Jewish professor in Göttingen described Teichmüller as ‘a very young, scientifically gifted man, but completely muddled and notoriously crazy’. One day, as Landau arrived at the lecture theatre, his path was blocked by the zealous young Nazi. Teichmüller told Landau that his Jewish way of presenting calculus was fundamentally incompatible with the Aryan way of thinking. Landau crumbled under the pressure, resigned his position and retired to Berlin. It hurt him greatly to be denied the chance to teach. Hardy invited him to Cambridge to give some lectures. ‘It was quite pathetic to see his delight when he found himself again in front of a blackboard and his sorrow when his opportunity came to an end,’ Hardy recalled. Unable to contemplate abandoning his homeland, Landau returned to Germany, where he died in 1938.
That year, Siegel – who had no Jewish connections – was moved from Frankfurt to Göttingen to try to rescue the reputation of the mathematics department. In 1940 he went into self-imposed exile in America to protest at the horrors of the war. After his terrible childhood experiences during the First World War, he had vowed never to remain in Germany if his country ever went to war again. He spent the war years at the Institute of Advanced Study in Princeton. Of the mathematicians who had forged Göttingen’s great reputation, only Hilbert remained in Germany. He had always been rather obsessed with Göttingen’s mathematical dominance. An old man now, he couldn’t comprehend the devastation that was happening around him. Siegel tried to explain to Hilbert why many members of the faculty had left. Siegel recalled, ‘I felt that he had the impression we were trying to play a bad joke on him.’
Within the space of a few weeks, Hitler had destroyed the great Göttingen tradition forged by Gauss, Riemann, Dirichlet and Hilbert. One commentator wrote that it was ‘one of the greatest tragedies experienced by human culture since the time of the Renaissance’. Göttingen (and, some might add, German mathematics itself) has never fully recovered from its destruction by Nazi Germany during the thirties. Hilbert died on St Valentine’s Day in 1943 after suffering a fall in Göttingen’s medieval streets. His death marked the end of the city’s position as the Mecca of mathematics.
Across the whole of Europe, mathematics was plunged into crisis. As nations prepared themselves for the inevitable confrontation, it became more and more impossible to justify the pursuit of abstract ideas for their own sake. Once again, European science became geared to giving nations a military edge. Many mathematicians would follow Siegel and emigrate from Europe to America. Most found the prosperity and support they encountered on the other side of the Atlantic the perfect environment for pure research. While America benefited from this academic migration, Europe has never regained its position as the world’s mathematical powerhouse.
Some mathematicians did return from exile. Once the war had finished, Siegel made his way back to Germany. Sheltered in Princeton, he had been totally cut off from mathematical developments in Europe and thought that little could have happened during his absence. He was to get a surprise. Whilst most mathematicians had fled or stopped doing mathematics, it turned out that there was indeed one piece of news. Siegel met up with his friend Harald Bohr, Hardy’s collaborator in attempts at the Riemann Hypothesis in Copenhagen. ‘So, has anything happened since my exile in Princeton?’ Siegel asked his old colleague. Bohr simply replied, ‘Selberg!’
Selberg, the solitary Scandinavian
In 1940 Siegel had made his way to Princeton via Norway. He had been invited to deliver a lecture at the University of Oslo. The German authorities approved the visit, ignorant of the fact that Siegel was using the lecture as a front. The principal purpose of the trip was to make his escape from Europe on a ship that was leaving Oslo bound for America. As the ship pulled out of the harbour, Siegel watched a fleet of German merchant ships approaching. Later he learnt that those ships were the advance party for the German invasion force. He had escaped, but left behind in the mathematics department at the University of Oslo was a young mathematician by the name of Atle Selberg. This young gun was burying his head in the mathematical sands trying to ignore the turmoil around him.
Even before war engulfed the region, Selberg was happy to spend his working days in self-imposed isolation. A cloistered existence often forces the mathematician to head off in a completely new direction. Selberg had already decided to work in a field of mathematics that no one else in the region was particularly familiar with. The fact that he had no help from colleagues in his mathematical endeavours did not deter him. Far from it – he seemed to revel in the isolation. As war approached and Norway became increasingly cut off, with foreign scientific journals failing to get through, Selberg found the silence an inspiration. ‘It was like being in a kind of prison. You were cut off. You certainly got the opportunity to concentrate on your own ideas. You were not distracted by what other people do. In that sense I considered that the situation in many ways was rather a good one for doing my work.’
This self-sufficiency was to characterise Selberg’s mathematical life. It had been cultivated during his youth, when he would sit undisturbed for hours in his father’s personal library delving into the many mathematics books on its shelves. It was during those long hours that Selberg came across an article about Ramanujan in a journal of the Norwegian Mathematical Society. Selberg recalls how the ‘strange and beautiful formulas … made a very deep and lasting impression on me’. The work of Ramanujan became one of Selberg’s main inspirations. ‘It seemed like a revelation – a completely new world to me, with much more appeal to the imagination.’ As a present, his father gave him Ramanujan’s Collected Papers, which Selberg still carries with him today. Self-taught with the help of his father’s large collection of mathematics books, he was already producing original work by the time he entered the University of Oslo in 1935.
He was particularly fascinated by Ramanujan’s formula for the sequence of partition numbers, which the Indian mathematician had discovered with Hardy. Although Ramanujan’s formula was recognised as a stunning achievement, there was something slightly unsatisfactory about it. The formula generated an answer which was not a whole number; the nearest whole number was the partition number itself. Surely there was a formula that produced exactly the number of partitions. Selberg was delighted when in the autumn of 1937 he succeeded in going one better than Ramanujan, coming up with an exact formula. Shortly after his discovery he was reading a review of his very first paper when his eyes were drawn to the review alongside it. To his great disappointment, he’d been beaten to the finishing line by Hans Rademacher, in a paper published the year before. Rademacher had fled his native Germany to America in 1934 after the Nazis terminated his employment at Breslau because of his pacifist sympathies. ‘I felt it was a bit of a blow at the time, later I got more used to such things!’ That Selberg had not heard of Rademacher’s contribution illustrates the degree to which Norway was now isolated from mathematical developments further afield.
It was always something of a surprise to Selberg that Hardy and Ramanujan had missed the exact formula. ‘I believe firmly that the responsibility for this rests with Hardy … Hardy did not fully trust Ramanujan’s insight and intuition … I think that if Hardy had trusted Ramanujan more, they should have inevitably ended with the Rademacher series. There is little doubt about that.’ Maybe, but the route they took did result in Hardy and Littlewood’s contribution to the Goldbach Conjecture, which otherwise might not have happened.
Selberg began to read as much as he could of the work of the Cambridge trio – Ramanujan, Hardy and Littlewood. He was particularly taken by their work on primes and connections to the zeta function. There was a statement in one of Hardy and Littlewood’s papers that especially intrigued him. They had written that their current methods seemed to offer no hope of proving that most zeros, the points at sea level in Riemann’s landscape, would be on Riemann’s ley line. Hardy had taken the great step of proving that there were at least an infinite number of zeros on the line, but he had failed to show that this infinite number amounted to even a fraction of the total number of zeros. Despite some improvements that he made with Littlewood, the number of zeros they could prove were on the line was swamped by the zeros they couldn’t catch. They stated boldly that their result could not be improved with the methods they had developed.
But Selberg was not as pessimistic as Hardy and Littlewood. He thought there was still some mileage to be had from their ideas. ‘I was looking at that passage in Hardy and Littlewood’s original paper where they explain at the end why their method would not give more than they could prove. I read that and I thought about it. And then I realised that what they had there was complete nonsense.’ Selberg’s hunch that he could go further than Hardy and Littlewood proved right. Although he still could not prove that all the zeros were on the line, he was able to show that the percentage captured by his method would not tail off to zero as he counted farther north. He wasn’t too sure what fraction of the total number of zeros he had caught, but this was the first substantial bite out of the pie which left some tooth marks. In retrospect, it looks as if he managed to prove that about 5 to 10 per cent of the zeros were on the line. As you counted north, then, at least this proportion of zeros obeyed the Riemann Hypothesis.
Even if it wasn’t a proof of the Riemann Hypothesis, Selberg’s bite was still a psychological breakthrough. But no one yet knew about it. Selberg himself wasn’t sure whether he might have been beaten to the result. The war ended, and he was invited to speak at the Scandinavian Congress of Mathematicians in Copenhagen in the summer of 1946. He had already been disappointed at being beaten to his discovery of the exact formula for the number of partitions, so he decided he had better check whether his result about the zeros was old news or not. But the University of Oslo had still not received copies of the journals which had failed to get through during the war. ‘I had heard that the library at the Institute of Technology in Trondheim had received copies. So I went up to Trondheim specifically for that. I spent about a week in the library.’
He needn’t have worried. He found himself way ahead of anyone else’s appreciation of the zeros in Riemann’s zeta landscape. His lecture in Copenhagen confirmed Bohr’s declaration to the visitors from America that the mathematical news in Europe amounted to ‘Selberg!’ Selberg spoke about his views on the Riemann Hypothesis. Although he had made a major contribution on the way to a proof, he stressed that there was still very little to support its truth. ‘I think the reason that we were tempted to believe the Riemann Hypothesis then was essentially that it is the most beautiful and simple distribution that we can have. You have this symmetry down the line. It would lead also to the most natural distribution of primes. You think that at least something should be right in this universe.’
Some misinterpreted his comments, thinking that Selberg was casting doubt on the validity of the Riemann Hypothesis. Yet he was not as pessimistic as Littlewood who believed the lack of evidence meant the Hypothesis was false. ‘I have always been a strong believer in the Riemann Hypothesis. I would never bet against it. But at that stage I maintained that we didn’t really have any results either numerical or theoretical that pointed very strongly to its truth. What the results pointed to was that it was mostly true.’ In other words, most zeros were probably on the line, just as Riemann had claimed he could prove nearly a century before.
Selberg’s wartime breakthrough was the death-rattle of European dominance in mathematics. Following his success he was headhunted by Hermann Weyl, a professor at the Institute for Advanced Study in Princeton, who in 1933 had escaped the worsening situation in Göttingen. This lone mathematician who had remained in Europe and endured the privations of the Second World War succumbed to the attractions that beckoned from the other side of the Atlantic. Selberg took up his invitation to visit the Institute, excited by the prospect of new inspiration. He arrived at the bustling port of New York and made his way to the sleepy town of Princeton, a short drive south of Manhattan.
The United States was to benefit immensely from the influx from overseas of talented mathematicians such as Selberg. Once a backwater of mathematical activity, America was now becoming the major power it still is today. It is the home of mathematics, drawing mathematicians from across the globe. Although Göttingen’s reputation as the Mecca of mathematics had been smashed by the devastation wrought by Hitler and the Second World War, it would rise phoenix-like at the Institute in Princeton.
The Institute had been founded in 1932 with the help of a five million dollar endowment from Louis Bamberger and his sister Caroline Bamberger Fuld. Its aim was to attract the world’s best scholars by offering them a peaceful haven and a handsome salary – indeed, the place gained the nickname of the Institute for Advanced Salaries. It sought to reproduce the collegiate atmosphere of Oxford and Cambridge where scholars from all disciplines could benefit by interacting with one another.
In contrast, however, to the musty atmosphere of those ancient seats of learning, Princeton had the air of somewhere young and fresh, bursting with life and ideas. Whilst it was considered a faux pas at Oxford or Cambridge to talk shop at high table, Princeton knew no such niceties. Members of the Institute talked openly about their work, whenever they wanted to. Einstein described it as a pipe yet unsmoked. ‘Princeton is a wondrous spot, a quaint and ceremonious village of puny demigods on stilts. Yet by ignoring certain social conventions I have been able to create for myself an atmosphere conducive to study and free from distraction. Into this small university town the chaotic voices of human strife barely penetrate.’
Although the Institute was founded to cater for all disciplines, it began life in the old mathematics building of Princeton University. The mathematics department would later move to the only skyscraper in Princeton and take its name – Fine Hall – with it. The Institute’s first home probably influenced its particular strengths, in mathematics and physics. Inscribed above the fireplace of the faculty lounge were some words which Einstein would often quote: ‘Raffiniert ist der Herr Gott, aber boshaft ist Er nicht’ (‘The Lord God is subtle, but malicious he is not’). The mathematicians, though, were rather more sceptical about the truth of such a statement. As Hardy had explained to Ramanujan, there is ‘diabolical malice inherent in the primes’.
The Institute moved to its new premises in 1940. Situated on the outskirts of Princeton and surrounded by woodland, it was insulated from the horrors happening in the outside world. Einstein described his exile to Princeton as ‘a banishment to paradise. I wished for this isolation all my life and now I have finally achieved it here in Princeton.’ In many ways the Institute would mirror its ancestor in Göttingen. It thrived on its isolation. People came from far and wide and were sucked into its self-sufficient community. Some would say that Princeton’s self-sufficiency grew into self-satisfaction. Not only had they accepted Göttingen’s mathematicians, but they appeared to have appropriated the German town’s motto: for members of the Institute, there was no life outside Princeton. Isolated in the woods, the Institute provided the perfect working environment for banished and fleeing Europeans.
Erdos, the wizard from Budapest
There was another mathematical émigré from Europe at the Institute whose life was to become intertwined with Selberg’s. While Ramanujan’s story had been inspiring the young Selberg in Norway, its magic was also working on another young mind. Paul Erdos, a Hungarian, was to become one of the most intriguing mathematicians of the second half of the twentieth century. But Ramanujan would not be the only thing to link these two young mathematicians. There was also controversy.
Whereas Selberg liked to work alone, Erdos thrived on collaboration. His stooped figure, clad in sandals and a suit, was familiar in mathematical common rooms across the world. He could be seen hunched over a notepad with a new collaborator at his side as they indulged his passion for creating and solving problems about numbers. He wrote over fifteen hundred papers in his lifetime, a phenomenal achievement. The only mathematician to have written more papers is Euler. Erdos was a mathematical monk who shed all his personal possessions lest they distract him from his mission. He gave away any money he earned to students or as rewards for answers to the many questions he posed. Like Hardy before him, God played a leading albeit unconventional role in his view of the world. The ‘Supreme Fascist’ was the name he gave to the custodian of the ‘Great Book’, which contained details of all the most elegant proofs of mathematical problems, both solved and unsolved. Erdos’s highest compliment for a proof was ‘that’s straight from the Book!’ He believed that all babies – or ‘epsilons’ as he called them, after the Greek letter that mathematicians use for a very small number – are born with knowledge of the Great Book’s proof of the Riemann Hypothesis. The trouble was that, after six months, they had forgotten it.
Erdos enjoyed doing his mathematics to music, and was often to be seen at concerts scribbling away in a notebook, unable to contain the excitement of a new idea. Although he was a great collaborator and hated to be alone, he found physical contact quite abhorrent. It was mental pleasure that sustained him, fuelled by a diet of coffee and caffeine tablets. As he once famously explained, ‘A mathematician is a machine for turning coffee into theorems.’
Erdos, like so many great mathematicians, was lucky to have a father who could expose him to ideas that would stimulate his passion for numbers. On one occasion his father had shown Erdos Euclid’s proof that there were infinitely many prime numbers. But Erdos was fascinated when his father twisted Euclid’s argument to prove that you could find stretches of numbers of arbitrary length where there were no primes.
If you want a sequence of 100 consecutive numbers where there are no primes, just take all the numbers up to 101 and multiply them together. The result is a number called the factorial of 101 and written as 101!. Then 101! is certainly divisible by all the numbers from 1 to 101. But if N is any of these numbers, then 101! + N will also be divisible by N, since 101! and N are both divisible by N. So all the numbers
are not prime. Here, then, is a list of 100 consecutive numbers, none of which are prime.
Erdos’s interest was piqued. How long would he have to count from 101! or some other number before he was guaranteed to find a prime number? Euclid had made sure there must be a prime somewhere, but would you have to wait an arbitrarily long time before finding the next prime? After all, if primes were being selected by the tossing of Nature’s coin, there’s no knowing how long it will be from one ‘heads’ to another. Of course, getting 1,000 tails in a row is very unlikely – but not impossible. As Erdos explored further, he learnt that in this respect the primes were not like the tossing of a coin. They may look a chaotic bunch of numbers, but their behaviour isn’t completely random.
In fact it was a French mathematician, Joseph Bertrand, who in 1845 first guessed how far you need to go before you’re guaranteed to find a prime. He believed that if you take any number, for example 1,009, and you count up to twice that number, then you should be guaranteed to find a prime on the way. There are actually quite a lot of primes between 1,009 and 2,018, the first being 1,013. Would this be true if Bertrand chose any number N? He couldn’t prove that you’ll always find a prime between any number N and its double, 2N. But the striking prediction he made, when he was only twenty-three, that this would always be the case became known as Bertrand’s Postulate.
It didn’t hold out as long Riemann’s Hypothesis as an unsolved problem. Within seven years the Russian mathematician Pafnuty Chebyshev had come up with a proof. Chebyshev used ideas similar to those he had employed in making the first inroads into the Prime Number Theorem, when he proved that Gauss’s guess was never more than 11 per cent away from the true number of primes. His methods were not as sophisticated as the powerful ones that Riemann had developed, but they were effective. So, unlike the tossing of a coin where there are no guarantees when the next head will appear, Chebyshev proved that there was a small measure of predictability to the primes.
One of the first results that Erdos published in 1931, when he was only eighteen, was a new proof of Bertrand’s Postulate. But to his dismay, someone pointed him to Ramanujan’s work and he discovered that his proof wasn’t as new as he had hoped. One of Ramanujan’s last achievements was an argument that greatly simplified Chebyshev’s proof of Bertrand’s Postulate. Although the young Erdos was rather upset, this was more than outweighed by the joy of discovering Ramanujan.
Erdos decided to see whether he could do better than Ramanujan and Chebyshev. He began to look at how big the gap between primes might be. The problem of the difference between primes was one that would continue to fascinate Erdos throughout his life. He was famous for offering prizes as rewards for proving his own conjectures. The second-largest he ever offered, $10,000, was for a proof of his conjecture about how big the gap between consecutive primes really is. The problem remains unsolved to this day, and the money is still there to be claimed, even though Erdos is no longer alive to appreciate the proof. But, as he liked to joke, the work that had to be done to earn one of his prizes probably violated the minimum wage law. He once rashly offered 10 billion factorial dollars for a proof of a conjecture which generalised Gauss’s Prime Number Theorem (10 billion factorial is the product of all the numbers from 1 to 10 billion). 100 factorial is already a number which exceeds the number of atoms in the universe, and Erdos expressed his relief when, in the 1960s, the mathematician who produced a proof didn’t claim his reward.
As soon as Erdos arrived at the Institute for Advanced Study in the late 1930s, he made his mark. Mark Kac was an émigré from Poland who was sheltering from the storm in Europe. Although Kac was interested in the theory of probability, he had announced a lecture which aroused Erdos’s interest. Kac was going to discuss a function that kept track of how many different primes divide each number as one counted higher. For example, 15 = 3 × 5 is divisible by two different prime numbers, whilst 16 = 2 × 2 × 2 × 2 is divisible by only one. So each number gets a score according to how many different primes divide into it.
Erdos recalled that Hardy and Ramanujan had been interested in how these scores varied. But it required a statistician like Kac to see that these scores were behaving in a completely random fashion. Kac could see that if you plotted a graph recording the scores made by each number as you counted higher, the shape of the graph was the familiar bell-shaped curve known to statisticians as the signature of randomness. Although Kac had recognised the behaviour of the function counting the number of prime building blocks, he didn’t possess the tricks from number theory that were needed to prove his hunch about this randomness. ‘I first stated the conjecture during a lecture in Princeton in March 1939. Fortunately for me and possibly for mathematics, Erdos was in the audience and he immediately perked up. Before the lecture was over he had completed the proof.’
This success began Erdos’s lifelong passion for mixing number theory and probability theory. At first sight the subjects look like chalk and cheese. Hardy once dismissively declared, ‘Probability is not a notion of pure mathematics but of philosophy or physics.’ The objects studied by number theorists have been set in stone since the beginning of time, immovable and unchanging. As Hardy said, 317 is a prime whether we like it or not. Probability theory, on the other hand, is the ultimate slippery subject. You’re never quite sure what’s going to happen next.
Orderly zeros mean random primes
Although Gauss had used the idea of tossing a prime number coin to guess at the number of primes, it wasn’t until the twentieth century that mathematicians were happy to contemplate a union of the diverse disciplines of probability and number theory. In the first few decades of the century, physicists were proposing that chance was an integral part of the subatomic world. An electron might behave as though it were a tiny billiard ball, but you can never be too sure where this ball is located. Many physicists were reluctant to admit it, but it seems that the role of a quantum dice dictates where you will find the electron. Perhaps the unsettling effect of the emerging theory of quantum physics and its probabilistic model of the world helped to challenge the view that chance had no role to play in something as deterministic as the primes. While Einstein was trying to deny that God played dice with Nature, down the corridor at the Institute Erdos was proving that the throw of the dice lay at the heart of number theory.
Indeed, it was during this period that mathematicians began to understand how the Riemann Hypothesis, which was about the regimented behaviour of the zeros in the zeta landscape, explained why the primes look so wild and random. The best way to understand this tension between the order of the zeros and the chaos of the primes is to take a deeper look at the quintessential model of randomness – the tossing of a coin.
If you toss a coin a million times, you should get half heads and half tails. But you wouldn’t expect to get an exact score. With a fair coin – one that behaves randomly, without any bias – you shouldn’t be surprised to see an error of about 1,000 either side of 500,000 heads. The theory of probability has provided a measure of how big this error can be if the experiment has its source in some random process. If the coin is tossed N times, there will be some deviation from N heads – the ‘error’ – one way or the other. This error has been analysed for a fair coin and is expected to be in the order of the square root of N. Thus, for example, out of 1,000,000 tosses of a fair coin, the number of heads is most likely to lie somewhere between 499,000 and 501,000 (1,000 being the square root of 1,000,000). If the coin were biased, you would expect the error to be consistently more than the square root of N.
Gauss had modelled his guess at the number of primes by tossing a coin. The probability that it would land heads on the Nth toss was only 1/log(N) rather than . However, in the same way that a conventional coin doesn’t come up exactly half heads, half tails, Nature’s prime number coin was not indicating exactly the number of primes that Gauss had predicted. But what was the error like? Was it within the limits for a coin behaving randomly, or was there a strong bias for producing primes in certain regions of numbers and leaving other regions barren?
The answer lies in the Riemann Hypothesis and its prediction about the location of the zeros. These points at sea level control the errors made by Gauss’s guess for the number of primes. Each zero with east—west coordinate equal to produces an error of N (which is another way of writing the square root of N). So if Riemann was correct about the location of the zeros, then the error between Gauss’s guess for the number of primes less than N and the true number of primes is at most of the order of the square root of N. This is the error margin expected by the theory of probability if the coin is fair, behaving randomly with no bias.
If the Riemann Hypothesis is false and there are zeros farther to the east of Riemann’s critical line, these zeros will produce an error which is much bigger than the square root of N. It would be like the coin producing many more heads than the fifty—fifty split expected from the fair coin. If the Riemann Hypothesis is false, that would imply that the prime number coin is far from fair. The farther east one finds zeros off Riemann’s ley line, the more biased is the prime number coin.
A fair coin produces truly random behaviour, whereas a biased coin produces a pattern. The Riemann Hypothesis therefore captures why the primes look so random. Riemann’s brilliant insight had turned this randomness on its head by finding the connection between the zeros of his landscape and the primes. To prove that the primes are truly random, one has to prove that on the other side of Riemann’s looking-glass the zeros are ordered along his critical line.
Erdos liked this probabilistic interpretation of the Riemann Hypothesis. For one thing, it reminded mathematicians why they had entered Riemann’s looking-glass world in the first place. Erdos wanted to encourage a return to what number theory was fundamentally about: numbers. It was striking that, ever since Riemann’s wormhole had opened up and sucked mathematicians through to a new world, fewer number theorists were talking about numbers. They were much more concerned with navigating the geometry of Riemann’s zeta landscape, on the lookout for points at sea level, than with talking about the primes themselves. Erdos initiated an about-turn, to studying primes for their own sake. He soon found out that he was not alone on this return journey.
Mathematical controversy
Although Selberg had been fascinated primarily with Riemann’s zeta landscape, at Princeton his interest was shifting away from the zeta function and becoming more directly focused on the primes themselves. His mathematical exodus to America was combined with a return to the solid side of Riemann’s looking-glass.
Since de la Vallée-Poussin and Hadamard’s proof of the Prime Number Theorem, mathematicians were frustrated at not being able to find an easier way to prove Gauss’s connection between logarithms and prime numbers. Was it only with highly sophisticated tools such as Riemann’s zeta function and this imaginary landscape that mathematicians would be able to prove Gauss’s estimate of the number of primes? Mathematicians were prepared to admit that such tools might be necessary to prove that the estimate was as good as Riemann’s Hypothesis would imply, namely that the error would always be as little as the square root of N, but they believed there had to be a simpler way to get the first rough estimate that Gauss had predicted. They had hoped they could extend Chebyshev’s elementary approach which proved that Gauss was at least within 11 per cent of the correct answer. But as time went by, and fifty years of attempts to find a simpler proof had failed, people were beginning to believe that the sophisticated tools that Riemann had introduced, and de la Vallée-Poussin and Hadamard had exploited, were simply unavoidable.
Hardy didn’t believe there was an elementary proof. Not that he didn’t wish for one; mathematicians constantly strive not only for proof but also simplicity. Hardy was simply becoming pessimistic and doubting that such a thing existed. He would have appreciated the contribution made by Erdos and Selberg who, just a few months after he died in 1947, found an elementary argument that linked primes and logarithms. However, the controversy that surrounded the credit for this elementary proof would have appalled him. The story has been told in various places, not least in two biographies of Erdos. Given the huge network of collaborators and correspondents that Erdos cultivated, combined with Selberg’s reticence, it is not surprising that the majority of these stories have been told from Erdos’s viewpoint. It is worth, however, recording something of Selberg’s side of the affair.
The first to wield the sophisticated tool of the zeta function was Dirichlet, who used it to confirm one of Fermat’s hunches. Dirichlet proved that if you take a clock calculator with N hours on the clock face, and you feed in the primes, the calculator will hit one o’clock infinitely often. In other words, there are infinitely many primes that have remainder 1 after dividing by N. Dirichlet’s proof had relied on the sophisticated use of the zeta function. His proof was the catalyst for Riemann’s great discoveries.
However, in 1946, nearly 110 years after Dirichlet’s discovery, Selberg came up with an elementary proof of Dirichlet’s Theorem that was closer in spirit to Euclid’s proof that there are infinitely many primes. His proof, avoiding the zeta function, was a psychological breakthrough at a time when many believed it was impossible to make any headway in the theory of prime numbers without using Riemann’s ideas. The proof, though subtle, required no sophisticated nineteenth-century mathematics and could possibly even have been understood by the ancient Greeks themselves.
Paul Turán, a Hungarian mathematician visiting the Institute at Princeton, had become friendly with Selberg during the time they spent together. He was also a good friend of Erdos. In fact, a joint paper that he had written with Erdos was the only ID he could produce when a Soviet military patrol stopped him in the streets of liberated Budapest in 1945. The patrol was suitably impressed, and Turán was saved from a trip to the gulag. As Turán later joked, this was ‘a surprising application of number theory’.
Turán was keen to understand something of the ideas behind Selberg’s proof of Dirichlet’s result, but he was due to leave the Institute after spending the spring there. Selberg was happy to show him some of the details, and even suggested that Turuán lecture on the proof while Selberg went to renew his visa during a trip to Canada. But during his discussion with Turán, Selberg showed slightly more of his hand than he had meant to.
During the lecture Turuán mentioned a rather extraordinary formula Selberg had proved, one not directly related to the proof of Dirichlet’s Theorem. Erdos was in the audience and saw that this formula was just what he needed to improve on Bertrand’s Postulate, that there will always be a prime between N and 2N. What Erdos was trying to do was to see whether you needed to go as far as 2 times N. For example, could you always find a prime between N and 1.01 times N? He realised that this wouldn’t work for every N. After all, if N is 100, there are no whole numbers, let alone prime numbers, between 100 and 101 (which is 100 times 1.01). But Erdos believed that once N was big enough, then, in the spirit of Bertrand’s Postulate, there would be a prime between N and 1.01N. There was nothing special about 1.01. Erdos believed this would work for any choice of number between 1 and 2. Having listened to Turán’s lecture, Erdos could see that Selberg’s formula provided the missing link in his proof.
‘Erdos asked me when I got back if I had anything against him using this to give an elementary proof of this generalisation of Bertrand’s Postulate.’ It was a result that Selberg had himself thought about, but he hadn’t got anywhere. ‘I was not working on that so I said that I had no objection.’ Selberg was distracted by a multitude of practical problems at the time. He had to renew his visa, find somewhere to live in Syracuse, where he had accepted a position for the coming academic year, and prepare lectures to teach a summer school for engineers. ‘At any rate, Erdos was always rather quick at things and he managed to find a proof.’
Now there were certain things that Selberg had not let on to Turuán. In particular, the reason Selberg had been thinking about this generalisation of Bertrand’s postulate was that he could see how to fit it into a jigsaw to complete the picture of an elementary proof of the Prime Number Theorem. With Erdos’s result, Selberg now had that final piece that gave him the proof.
He told Erdos how he had used his result to complete an elementary proof of the Prime Number Theorem. Erdos suggested they present the work to the small group that had been present at Turuán’s lecture. But Erdos couldn’t contain his excitement, and he busily began issuing invitations to what he promised would be a very interesting lecture. Selberg had not expected such a large audience.
When I arrived there in the late afternoon, around 4 or 5, the room was packed. So I went up and I went through the argument and then I asked Erdos to go through his part. Then I went through the rest that was needed to complete the proof. So the first proof was obtained by using this intermediary result that he had got.
Erdos proposed that they write a paper together explaining the proof. But as Selberg explains,
I had never published joint papers. I really wanted to publish separate papers but Erdos insisted that one should do things the same way that Hardy and Littlewood had done things. But I had never agreed to cooperate. When I came to the States I had done all my mathematics in Norway. It was done alone, even without talking to anybody … no, I have never been a collaborator in that sense. I talk with people but I work alone, this is what suits my temperament.
The truth is that here were two mathematicians with completely different temperaments. One was an entirely self-sufficient loner who wrote only one joint paper in his life, with the Indian mathematician Saravadam Chowla, and that somewhat against his will. The other took collaboration to such an extreme that mathematicians talk of their Erdos number, the number of co-authors that link them to a paper with Erdos. Mine is 3, which means I’ve written a paper with someone who’s written a paper with someone who’s written a paper with Erdos. Since Chowla was one of Erdos’s 507 co-authors, Selberg’s one joint paper that he ever wrote gave him an Erdos number of 2. Over five thousand mathematicians have an Erdos number of 2.
After this refusal, as Selberg admits, ‘things got out of hand’. By 1947 Erdos had built up an extensive network of collaborators and correspondents. He would keep them up to date with his mathematical progress by firing off postcards. The story goes that the nail in the coffin for Selberg was being greeted on his arrival in Syracuse by a faculty member who asked, ‘Have you heard the news? Erdos and some Scandinavian mathematician have produced an elementary proof of the Prime Number Theorem.’ By then Selberg had found an alternative argument that avoided the need for the intermediary step that Erdos had provided him. Selberg went ahead and published alone. His paper appeared in the Annals of Mathematics, the Princeton-based publication generally regarded as one of the three leading mathematical journals in the world, and where Andrew Wiles eventually published his proof of Fermat’s Last Theorem.
Erdos was furious. He asked Hermann Weyl to adjudicate the issue. Selberg recounts, ‘I take pleasure in the fact that Hermann Weyl essentially came down on my side in the end after he had heard both sides.’ Erdos published his proof acknowledging Selberg’s role. But it was a very unfortunate episode. Despite the unworldly nature of mathematics, mathematicians still have egos that need massaging. Nothing acts as a better drive to the creative process than the thought of the immortality bestowed by having your name attached to a theorem. The story of Selberg and Erdos highlights the importance in mathematics – indeed, in all of science – of credit and priority. That is why Wiles spent seven years alone in his attic working on Fermat’s Last Theorem in secret, lest he have to share the glory.
Although mathematicians are like runners in a relay team, passing the baton from one generation to the next, they still yearn all the while for the individual glory that crossing the finishing line will bestow. Mathematical research is a complex balance between the need for collaboration in projects which can span centuries, and the longing for immortality.
After a while it became clear that Selberg’s elementary proof of the Prime Number Theorem wasn’t the striking breakthrough that had been hoped for. Some thought that the insight might provide an elementary path to proving the Riemann Hypothesis. After all, it might have shown that the difference between Gauss’s guess and the actual number of primes would never be more than the square root of N off its mark. And people knew that this was equivalent to the zeros falling onto Riemann’s regimented straight line.
By the end of the 1940s, Selberg still held the record for proving how many zeros lay on Riemann’s line. That was one of the achievements for which he was awarded a Fields Medal in 1950. Hadamard, who was then eighty, was due to attend the International Congress of Mathematicians in Cambridge, Massachusetts, to celebrate Selberg’s award. He was particularly looking forward to meeting the explorer who’d found an elementary route to the base camp that he and de la Vallée-Poussin had established fifty years before. However, both Hadamard and Laurent Schwartz, the other mathematician due to receive a Fields Medal, were denied visas because of their Soviet connections – McCarthyism had just begun to raise its ugly head. It required President Truman’s intervention before they were allowed to enter America, just days before the congress.
People have subsequently extended Selberg’s arguments about the percentage of zeros that we can prove are really on Riemann’s ley line, adding their own ingenious twists. Some proofs of mathematical theorems evolve very naturally once you have an idea of the general direction in which to head. Finding the first bit of the path is the hard part. Improving Selberg’s estimate is very different, however. The proofs require very delicate analysis. They are not susceptible to one great idea but require tremendous perseverance to see them through to their end. The path is littered with traps. One false move, and a number you thought was bigger than zero can suddenly turn negative on you. Each step needs to be taken with great care, and mistakes can easily creep in.
In the 1970s, Norman Levinson improved on Selberg’s estimates and thought at one stage that he’d managed to capture as many as 98.6 per cent of the zeros. Levinson gave his colleague Gian-Carlo Rota at MIT a copy of the manuscript of the proof, joking that he’d proved that 100 per cent were on the line – the manuscript did 98.6 per cent of the zeros, and the other 1.4 per cent were left to the reader. Rota thought he was serious, and started spreading the word that Levinson had proved the Riemann Hypothesis. Of course, even if he had got to 100 per cent, that did not necessarily mean that all the zeros were on the line because we are dealing with the infinite. But that did not stop the rumour spreading.
Eventually a mistake was found in the manuscript which brought the located zeros down to 34 per cent. Still, this was a record that stood for some time, and was all the more impressive because Levinson was in his sixties when he did his best work. As Selberg says, ‘He had to have a great deal of courage to carry out such numerical calculation because in advance you wouldn’t know whether it would lead anywhere.’ It was said that Levinson had great ideas for how to extend his methods, but he died from a brain tumour before he could bring them to fruition. The record currently belongs to Brian Conrey of Oklahoma University, who proved in 1987 that 40 per cent of the zeros must lie on the line. Conrey has some ideas about how to improve on his estimate, but the huge amount of work it would take doesn’t seem worth it for the extra few per cent. ‘It would be worth it if I could get the estimate above 50 per cent because then at least you could say most of the zeros were on the line.’
Erdos was very hurt by the controversy surrounding credit for the elementary proof, but he remained prolific throughout his life, defying the myths of ageing and mathematical burn-out. When he failed to secure a permanent position at Princeton, he chose instead the life of the itinerant mathematician. With no home and no job, he preferred to descend suddenly on one of his many friends around the world to indulge his love for collaboration, often staying with them for several weeks before moving on just as suddenly. He died in 1996, the centenary year of the first proof of the Prime Number Theorem. Erdos was still collaborating on joint papers at the age of eighty-three. He said, shortly before he died, it will be another million years, at least, before we understand the primes.’
Now silver-haired and in his nineties, Selberg is still reading the latest about the Riemann Hypothesis and attending conferences, at which he offers pearls of wisdom to young delegates. In his gentle voice you can still hear the singing tones of his home of Norway, but underneath are often penetrating and cutting commentaries on the work he is appraising. He does not suffer fools gladly. In 1996 his talk at a meeting in Seattle celebrating the centenary of the proof of the Prime Number Theorem was greeted with a standing ovation by six hundred mathematicians.
Selberg believes that despite much progress, we still have no real idea how to prove the Hypothesis:
I think it is anybody’s guess whether we are close to a solution or not. There are some people who think we are getting closer. Of course as time progresses if we ever get a solution then we are getting closer. But some believe that we have very essential elements of a solution. I don’t really see that. It’s very different from Fermat. There has been no corresponding breakthrough. It may very well survive a bicentennial by 2059 but of course I will not see that. How long the problem will last it is impossible to say. I do think that a solution will eventually be found. I don’t think it is a result which is unprovable. Maybe though the proof will be so involved that the human brain will not catch up with it.
In the lecture he delivered in Copenhagen after the war, Selberg had cast doubt on whether there was any evidence that the Riemann Hypothesis was true. Then it seemed like wishful thinking, but today his view has changed. The evidence that has emerged in the fifty years since the war has in Selberg’s view become quite overwhelming. But it was the war, and in particular the code-breakers at Bletchley Park, that were responsible for the development of the machine that would generate this new evidence: the computer.