The Music of the Primes: Why an unsolved problem in mathematics matters. Marcus Sautoy du

up a friendship was the eminent physicist Wilhelm Weber. Weber had collaborated on numerous projects with Gauss during their time together in Göttingen. They became a scientific Sherlock Holmes and Dr Watson, Gauss providing the theoretical underpinning which Weber then put into practice. One of their most famous inventions was to realise the potential of electromagnetism to communicate over a distance. They successfully rigged up a telegraph line between Gauss’s observatory and Weber’s laboratory over which they exchanged messages.

      Although Gauss thought the invention a curiosity, Weber saw clearly what their discovery would unleash. ‘When the globe is covered with a net of railroads and telegraph wires,’ he wrote, ‘this net will render services comparable to those of the nervous system in the human body, partly as a means of transport, partly as a means for the propagation of ideas and sensations with the speed of lightning.’ The rapid spread of the telegraph, together with the later implementation of Gauss’s invention of the clock calculator in computer security, make Gauss and Weber the grandfathers of e-business and the Internet. Their collaboration is immortalised in a statue of the pair in the city of Göttingen.

      One visitor to Weber in Göttingen paints a typical picture of a slightly mad inventor: ‘a curious little fellow who speaks in a shrill, unpleasant and hesitating voice. He speaks and stutters unceasingly; one has nothing to do but to listen. Sometimes he laughs for no earthly reason, and one feels sorry at not being able to join him.’ Weber had a little more of the rebel in him than his collaborator Gauss. He had been one of the ‘Göttingen Seven’ temporarily dismissed from the faculty for protesting at the arbitrary rule of the Hanoverian king in 1837. For some time after completing his thesis, Riemann worked as Weber’s assistant. During this apprenticeship Riemann developed a soft spot for Weber’s daughter, but his advances were not reciprocated.

      In 1854, Riemann wrote to his father that ‘Gauss is seriously ill and the physicians fear that his death is imminent.’ Riemann was worried that Gauss might die before examining his habilitation, the degree required to become a professor at a German university. Fortunately Gauss lived long enough to hear Riemann’s ideas on geometry and its relationship to physics that had germinated during his work with Weber. Riemann was convinced that the fundamental questions of physics could all be answered using mathematics alone. The developments in physics over the ensuing years would eventually confirm his faith in mathematics. Riemann’s theory of geometry is regarded by many as one of his most significant contributions to science, and it would be one of the planks in the platform from which Einstein launched his scientific revolution at the beginning of the twentieth century.

      A year later, Gauss died. Although the man had passed, his ideas were to keep mathematicians busy for generations to come. He had left behind his conjectured connection between primes and the logarithm function for the next generation to chew over. Astronomers immortalised the great man in the heavens by naming an asteroid Gaussia. And in the University of Göttingen’s anatomical collection one can even find Gauss’s brain, pickled for eternity, which was reported to be more richly convoluted than any brain previously dissected.

      Dirichlet, whose lectures Riemann had attended in Berlin, was appointed to Gauss’s vacant chair. Dirichlet was to bring to Göttingen some of the intellectual excitement that Riemann had enjoyed when he was in Berlin. An English mathematician recorded the impression that a visit to Dirichlet in Göttingen made on him at this time: ‘He is rather tall, lanky-looking man with moustache and beard about to turn grey … with a somewhat harsh voice and rather deaf: it was early, he was unwashed and unshaved and with his schlafrock [dressing gown], slippers, cup of coffee and cigar.’ Despite this Bohemian exterior, there burned inside him a desire for rigour and proof that was unequalled at the time. His contemporary in Berlin, Carl Jacobi, wrote to Dirichlet’s first patron Alexander von Humboldt that ‘Only Dirichlet, not I, nor Cauchy, not Gauss, knows what a perfectly rigorous proof is, but we learn it only from him. When Gauss says he has proved something, I think it is very likely; when Cauchy says it, it is a fifty-fifty bet; when Dirichlet says it, it is certain.’

      The arrival of Dirichlet in Göttingen began to shake the social fabric of the town. Dirichlet’s wife, Rebecka, was the sister of the composer Felix Mendelssohn. Rebecka loathed the dull Göttingen social scene and threw numerous parties trying to reproduce the Berlin salon atmosphere she had been forced to leave behind.

      Dirichlet’s less formal approach to the educational hierarchy meant that Riemann was able to discuss mathematics openly with the new professor. Riemann had become rather isolated on his return from Berlin to Göttingen. The combination of Gauss’s austere personality in later life and Riemann’s shyness meant that Riemann had discussed little with the great master. By contrast, Dirichlet’s relaxed manner was perfect for Riemann who, in an atmosphere more conducive to discussion, began to open up. Riemann wrote to his father about his new mentor: ‘Next morning Dirichlet was with me for two hours. He read over my dissertation and was very friendly – which I could hardly have expected considering the great distance in rank between us.’

      In turn, Dirichlet appreciated Riemann’s modesty and also recognised the originality of his work. On occasions Dirichlet even managed to drag Riemann away from the library to join him on walks in the countryside around Göttingen. Almost apologetically, Riemann wrote to his father explaining that these escapes from mathematics did him more good scientifically than if he had stayed at home poring over his books. It was during one of his discussions with Riemann whilst walking through the woods of Lower Saxony that Dirichlet inspired Riemann’s next move. It would open up a whole new perspective on the primes.

      The zeta function – the dialogue between music and mathematics

      During his years in Paris in the 1820s, Dirichlet had become fascinated by Gauss’s great youthful treatise Disquisitiones Arithmeticae. Although Gauss’s book marked a beginning of number theory as an independent discipline, the book was difficult and many failed to penetrate the concise style Gauss preferred. Dirichlet, though, was more than happy to battle with one tough paragraph after another. At night he would place the book under his pillow in the hope that the next morning’s reading would suddenly make sense. Gauss’s treatise has been described as a ‘book of seven seals’, but thanks to the labours and dreams of Dirichlet, those seals were broken and the treasures within gained the wide distribution they deserved.

      Dirichlet was especially interested in Gauss’s clock calculator. In particular, he was intrigued by a conjecture that went back to a pattern spotted by Fermat. If you took a clock calculator with N hours on it and you fed in the primes, then, Fermat conjectured, infinitely often the clock would hit one o’clock. So, for example, if you take a clock with 4 hours there are infinitely many primes which Fermat predicted would leave remainder 1 on division by 4. The list begins 5, 13, 17, 29, …

      In 1838, at the age of thirty-three, Dirichlet had made his mark in the theory of numbers by proving that Fermat’s hunch was indeed correct. He did this by mixing ideas from several areas of mathematics that didn’t look as if they had anything to do with one another. Instead of an elementary argument like Euclid’s cunning proof that there are infinitely many primes, Dirichlet used a sophisticated function that had first appeared on the mathematical circuit in Euler’s day. It was called the zeta function, and was denoted by the Greek letter ζ. The following equation provided Dirichlet with the rule for calculating the value of the zeta function when fed with a number x:

      To calculate the output at x, Dirichlet needed to carry out three mathematical steps. First, calculate the exponential numbers 1^x, 2^x, 3^x, …, n^x, … Then take the reciprocals of all the numbers produced in the first step. (The reciprocal of 2^x is 1/2^x.) Finally, add together all the answers from the second step.

      It is a complicated recipe. The fact that each number 1, 2, 3, … makes a contribution to the definition of the zeta function hints at its usefulness to the number theorist. The downside comes in having to deal with an infinite sum of numbers. Few mathematicians could have predicted