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

heard the first big theme in the music of the primes. But it was one of his few students, Riemann, who would truly unleash the full force of the hidden harmonies that lay behind the cacophony of the primes.

       CHAPTER THREE

       Riemann’s Imaginary Mathematical Looking-Glass

      Do you not feel and hear it? Do I alone hear this melody so wondrously and gently sounding … Richard Wagner. Tristan und Isolde (Act III, Scene iii)

      In 1809 Wilhelm von Humboldt became the education minister for the north German state of Prussia. In a letter to Goethe in 1816 he wrote, ‘I have busied myself here with science a great deal, but I have deeply felt the power antiquity has always wielded over me. The new disgusts me …’ Humboldt favoured a movement away from science as a means to an end, and a return to a more classical tradition of the pursuit of knowledge for its own sake. Previous education schemes had been geared to providing civil servants for the greater glory of Prussia. From now on, more emphasis was to be placed on education serving the needs of the individual rather than the state.

      In his role as a thinker and civil servant, Humboldt enacted a revolution with far-reaching effects. New schools, called Gymnasiums, were created across Prussia and the neighbouring state of Hanover. Eventually the teachers in these schools were not to be members of the clergy, as in the old education system, but graduates of the new universities and polytechnics that were built during this period.

      The jewel in the crown was Berlin University, founded in 1810 during the French occupation. Humboldt called it the ‘mother of all modern universities’. Housed in what had once been the palace of Prince Heinrich of Prussia on the grand boulevard Unter den Linden, the university would for the first time promote research alongside teaching. ‘University teaching not only enables an understanding of the unity of science but also its furtherance,’ Humboldt declared. Despite his passion for the Ancient World, under his guidance the university pioneered the introduction of new disciplines to sit beside the classical faculties of Law, Medicine, Philosophy and Theology.

      For the first time, the study of mathematics was to form a major part of the curriculum in the new Gymnasiums and universities. Students were encouraged to study mathematics for its own sake and not simply as a servant of the other sciences. This contrasted starkly with Napoleon’s educational reforms, which saw mathematics harnessed to further French military aims. Carl Jacobi, one of the professors in Berlin, wrote to Legendre in Paris in 1830 about the French mathematician Joseph Fourier, who had reproached the German school of thought for ignoring more practical problems:

      It is true that Fourier was of the opinion that the principal object of mathematics is public use and the explanation of natural phenomena; but a philosopher like him ought to have known that the sole object of the science is the honour of the human spirit, and that on this view a problem in the theory of numbers is worth as much as a problem of the system of the world.

      For Napoleon, it was education that would finally destroy the arcane rules of the ancien régime. His recognition that education was the backbone for building his new France had led to the establishment in Paris of some of the institutes which are still famous today. Not only were the colleges meritocratic, allowing students from all backgrounds to attend, but also the educational philosophy put a greater emphasis on education and science serving society. One of the French Revolutionary regional officers wrote to a professor of mathematics in 1794, commending him on teaching a course in ‘Republican arithmetic’: ‘Citizen. The Revolution not only improves our morals and paves the way for our happiness and that of future generations, it even unlooses the shackles that hold back scientific progress.’

      Humboldt’s approach to mathematics was very different from this utilitarian philosophy that prevailed across the border. The liberating effect of Germany’s educational revolution was to have a great impact on mathematicians’ understanding of many aspects of their field. It would allow them to establish a new, more abstract language of mathematics. In particular, it would revolutionise the study of prime numbers.

      One town that benefited from Humboldt’s initiatives was Lüneberg, in Hanover. Lüneberg, once a thriving commercial centre, was now a town in decline. Its narrow streets paved with cobblestones were no longer buzzing with the business it had seen in previous centuries. But in 1829 a new building was erected amidst the tall towers of the three Gothic churches in Lüneberg: the Gymnasium Johanneum.

      By the early 1840s the new school was flourishing. Its director, Schmalfuss, was a keen proponent of the neo-humanist ideals initiated by Humboldt. His library reflected his enlightened views: it featured not only the classics and the works of modern German writers, but also volumes from farther afield. In particular, Schmalfuss managed to get his hands on books coming out of Paris, the powerhouse of European intellectual activity during the first half of the century.

      Schmalfuss had just accepted a new boy at the Gymnasium Johanneum, Bernhard Riemann. Riemann was very shy and found it difficult to make friends. He had been attending the Gymnasium in the town of Hanover, where he had been boarding with his grandmother, but when she died, in 1842, he was forced to move to Lüneberg where he could board with one of the teachers. Joining the school after all his contemporaries had established friendships did not make life easy for Riemann. He was desperately homesick and was teased by the other children. He would rather walk the long distance back to his father’s house in Quickborn than play with his contemporaries.

      Riemann’s father, the pastor in Quickborn, had high expectations for his son. Although Bernhard was unhappy at school, he worked hard and conscientiously, determined not to disappoint his father. But he had to battle with an almost disabling streak of perfectionism. His teachers would often get frustrated at Riemann’s inability to submit his work. Unless it was perfect, the boy could not bear to suffer the indignity of anything less than full marks. His teachers began to doubt whether Riemann would ever be able to pass his final examinations.

      It was Schmalfuss who saw a way to bring the young boy on and exploit his obsession with perfection. Early on, Schmalfuss had spotted Riemann’s special mathematical skills and was keen to stimulate the student’s abilities. He allowed Riemann the freedom of his library, with its fine collection of books on mathematics, where the boy could escape the social pressures of his classmates. The library opened up a whole new world for Riemann, a place where he felt at home and in control. Suddenly here he was in a perfect, idealised mathematical world where proof prevented any collapse of this new world around him, and numbers became his friends.

      Humboldt’s drive from teaching science as a practical tool to the more aesthetic notion of knowledge for its own sake had filtered down to Schmalfuss’s classroom. The teacher steered Riemann’s reading away from mathematical texts full of formulas and rules that were aimed at feeding the demands of a growing industrial world, and guided him towards the classics of Euclid, Archimedes and Apollonius. With their geometry, the ancient Greeks sought to understand the abstract structure of points and lines, and they were not hung up on the particular formulas behind the geometry. When Schmalfuss did give Riemann a more modern text, Descartes’s treatise on analytical geometry – a subject rife with equations and formulas – the teacher could see that the mechanical method developed in the book did not appeal to Riemann’s growing taste for conceptual mathematics. As Schmalfuss later recalled in a letter to a friend, ‘already at that time he was a mathematician next to whose wealth a teacher felt poor’.

      One of the books that was sitting on Schmalfuss’s shelf was a contemporary volume the teacher had acquired from France. Published in 1808, Théorie des Nombres by Adrien-Marie Legendre was the first text to record the observation that there seemed to be a strange connection between the function that counted the number of prime numbers and the logarithm function. This connection, discovered by Gauss and Legendre, was only based on experimental evidence. It was far from clear whether, as one counted higher, the number of primes would always be approximated by Gauss’s or Legendre’s function.