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

map there was a point corresponding to every possible number. The imaginary numbers then simply became coordinates on a map. The number 1 + 2i was represented by the point reached by travelling one unit east and two units north.

      Gauss would interpret these numbers as sets of directions in his map of the imaginary world. To add two imaginary numbers A + Bi and C + Di just meant following two sets of directions, one after the other. For example, adding together 6 + 3i and 1 + 2i gets you to the location 7 + 5i.

      Following directions – how to add two imaginary numbers.

      Although this was a very potent picture. Gauss was to keep his map of the imaginary world hidden from public view. Once he had built his proof, he removed the graphic scaffolding so that no trace of his vision remained. He was aware that pictures in mathematics were regarded with some suspicion during this period. The dominance of the French mathematical tradition during Gauss’s youth meant that the preferred pathway to the mathematical world was the language of formulas and equations, a language that went hand in hand with the utilitarian approach to the subject. There were also other reasons for this aversion to images.

      For several hundred years, mathematicians had believed that pictures had the power to mislead. After all, the language of mathematics had been introduced to tame the physical world. In the seventeenth century, Descartes had attempted to turn the study of geometry into pure statements about numbers and equations. ‘Sense perceptions are sense deceptions’ was his motto. Riemann had come to dislike this denial of the physical picture when he’d been reading Descartes in the comfort of Schmalfuss’s library.

      Mathematicians around the turn of the nineteenth century had been burnt by an erroneous pictorial proof of a formula describing the relationship between the number of corners, edges and faces of geometric solids. Euler had conjectured that if a solid has C corners, E edges and F faces, then the numbers C, E and F must satisfy the relationship C − E + F = 2. For example, a cube has 8 corners, 12 edges and 6 faces. The young Cauchy had himself constructed a ‘proof’ in 1811 based on pictorial intuition, but was rather shocked to be shown a solid which didn’t satisfy this formula – a cube with a hole at its centre.

      The ‘proof’ had missed the possibility that a solid might contain such a hole. It was necessary to introduce an extra ingredient into the formula which kept track of the number of holes in the solid. Having been tricked by the power of pictures to hide perspectives that weren’t initially apparent, Cauchy sought refuge in the security that formulas seemed to provide. One of the revolutions he effected was to create a new mathematical language which allowed mathematicians to discuss the concept of symmetry in a rigorous way without the need for pictures.

      Gauss knew that his secret map of imaginary numbers would be anathema to mathematicians at the end of the eighteenth century, so he omitted it from his proof. Numbers were things you added and multiplied, not drew pictures of. Gauss eventually came clean some forty years later about the scaffolding he had used in his doctorate.

      Looking-glass world

      Even without Gauss’s map, Cauchy and other mathematicians had begun to explore what happens if you extend the idea of functions to this new world of imaginary numbers rather than sticking to real numbers. To their surprise, these imaginary numbers opened up new connections between seemingly unrelated parts of the mathematical world.

      A function is like a computer program in which you input one number, a calculation is made, and another number is output. The function might be defined by some simple equation like x² + 1. When you input a number, for example 2, the function calculates 2² + 1 and outputs 5. Other functions are more complicated. Gauss was interested in the function that counted the number of primes. You input a number x, and the function tells you how many primes there are up to x. Gauss had denoted this function by π(x). The graph of this function is a climbing staircase, as shown on p. 50. Every time the input encounters a prime number, the output jumps another step. As x goes from 4.9 to 5.1, the number of primes increases from two to three to register the new prime, 5.

      Mathematicians soon realised that some of these functions, such as the one built from the equation x² + 1, could be fed with imaginary numbers as well as ordinary numbers. For example, input x = 2i into the function, and the output is calculated as (2i)² + 1 = −4 + 1 = −3. The feeding of functions with imaginary numbers had begun in Euler’s generation. As early as 1748, Euler had stumbled across strange connections between unrelated bits of mathematics by taking a trip through this looking-glass world. Euler knew that when you fed the exponential function 2^x with ordinary numbers x, you got a graph that climbed rapidly. But when he fed the function with imaginary numbers he got a rather unexpected answer. Instead of the exponentially climbing graph, he started to see undulating waves of the type we now associate with, for example, sound waves. The function that produces these undulating waves is called the sine function. The image of the sine function is a familiar repeating curve where every 360 degrees we see the same shape appearing. The sine function is now used in a host of everyday calculations. For example, it can be used to measure the height of a building from ground level by measuring angles. It was Euler’s generation who discovered that these sine waves were also the key to reproducing musical sounds. A pure note like the A produced by a tuning fork used to tune a piano can be represented by such a wave.

      Euler fed imaginary numbers into the function 2^x. To his surprise, out came waves which corresponded to a particular musical note. Euler showed that the character of each note depended on the coordinates of the corresponding imaginary number. The farther north one is, the higher the pitch. The farther east, the louder the volume. Euler’s discovery was the first inkling that these imaginary numbers might open up unexpected new paths through the mathematical landscape. Following Euler, mathematicians began travelling out to this new-found land of imaginary numbers. The search for new connections would become infectious.

      Riemann returned to Göttingen in 1849 in order to complete his doctoral thesis for Gauss’s consideration. That was the year in which Gauss wrote to his friend Encke of his childhood discovery of the connection between primes and logarithms. Although Gauss probably discussed his discovery with members of the faculty in Göttingen, prime numbers were not yet on Riemann’s mind. He was buzzing with the new mathematics from Paris, keen to explore the strange emerging world of functions fed with imaginary numbers.

      Cauchy had begun the task of making a rigorous subject out of Euler’s first tentative steps into this new territory. Whilst the French were masters of equations and formulaic manipulation, Riemann was ready to capitalise on the German education system’s return to a more conceptual view of the world. By November 1851, his ideas had crystallised and he submitted his dissertation to the faculty in Göttingen. His ideas obviously struck a chord with Gauss. He greeted Riemann’s doctorate as evidence ‘of a creative, active, truly mathematical mind, and of a gloriously fertile originality’.

      Riemann wrote to his father, keen to tell him about the progress he was making: ‘I believe I have improved my prospects with my dissertation. I hope also to learn to write more quickly and more fluently in time, especially if I mingle in society.’ But academic life in Göttingen did not at first live up to the thrills of Berlin. It was a somewhat stuffy, insular university, and Riemann lacked the confidence to engage with the old intellectual hierarchy. There were fewer students in Göttingen with whom he could relate. He was mistrustful of other people and never really at ease in a social environment. ‘He has done the strangest things here only because he believes that nobody can bear him,’ wrote his contemporary Richard Dedekind. Riemann was a hypochondriac and susceptible to bouts of depression. He hid his face behind the security of an increasingly large black beard. He was extremely anxious about his finances, surviving on the uncertainty of half a dozen voluntary students’ fees. The workload he undertook combined with the pressures of poverty led to a temporary breakdown in 1854. However, his mood would lighten whenever Dirichlet, the star of the Berlin mathematical tradition, visited Göttingen.