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

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      Gauss broke from the past by stressing the value of proof. For him, presenting proofs was the primary goal of the mathematician, an ethos which has remained fundamental to this day. Without a proof of the connection between logarithms and primes, Gauss’s discovery was worthless to him. The freedom that the patronage of the Duke of Brunswick permitted him meant he could be quite choosy, almost indulgent about the mathematics he produced. His prime motivation was not fame and recognition but a personal understanding of the subject he loved. His seal bore the motto Pauca sed matura (‘Few but ripe’). Unless a result had reached full maturity it remained an entry in his diary or a doodle at the back of his table of logarithms.

      For Gauss, mathematics was a private pursuit. He even encrypted entries in his diary using his own secret language. Some of them are easy to unravel. For example, on July 10, 1796, Gauss wrote Archimedes’ famous declaration ‘Eureka!’ followed by the equation

      num = Δ + Δ + Δ

      which represents his discovery that every number can be written as the sum of three numbers from the list of triangular numbers, 1, 3, 6, 10, 15, 21, 28, …, those numbers for which Gauss had produced his schoolroom formula. For example, 50 = 1 + 21 + 28. But other entries remain a complete mystery. No one has been able to unravel what Gauss meant when he wrote on October 11, 1796, ‘Vicimus GEGAN’. Some have blamed Gauss’s failure to disseminate his discoveries for holding back the development of mathematics by half a century. If he had bothered to explain half of what he had discovered and not been so cryptic in the explanations he did offer, mathematics might have advanced at a quicker pace.

      Some people believe that Gauss kept his results to himself because the Paris Academy had rejected his great treatise on number theory, Disquisitiones Arithmeticae, as obscure and dense. Having been stung by rejection, to protect himself from any further humiliation he insisted that every last piece of the mathematical jigsaw be in place before he would consider publishing anything. Disquisitiones Arithmeticae did not receive immediate acclaim partly because Gauss continued to be cryptic even in the work he did expose to public view. He always claimed that mathematics was like a piece of architecture. An architect never leaves the scaffolding for people to see how the building was constructed. This was not a philosophy that helped mathematicians to penetrate Gauss’s mathematics.

      But there were other reasons why Paris was not as receptive to Gauss’s ideas as he had hoped. By the end of the eighteenth century, mathematics in Paris was becoming ever more dedicated to serving the demands of an increasingly industrialised state. The Revolution of 1789 had shown Napoleon the need for a more centralised teaching of military engineering, and he responded by founding the École Polytechnique to further his war aims. ‘The advancement and perfection of mathematics are intimately connected with the prosperity of the State,’ Napoleon declared. French mathematics was dedicated to solving problems of ballistics and hydraulics. But despite this emphasis on the practical needs of the state, Paris still boasted some of the leading pure mathematicians in Europe.

      One of the great authorities in Paris was Adrien-Marie Legendre, who was born twenty-five years before Gauss. Portraits of Legendre depict a rather puffed-up gentleman with a round, chubby face. In contrast to Gauss, Legendre came from a wealthy family but he had lost his fortune during the Revolution and been forced to rely instead on his mathematical talents for his livelihood. He too was interested in the primes and number theory, and in 1798, six years after Gauss’s childhood calculations, he announced his discovery of the experimental connection between primes and logarithms.

      Although it would later be proved that Gauss had indeed beaten Legendre to the discovery, Legendre did nonetheless improve on the estimate for the number of primes up to N. Gauss had guessed that there were roughly N/log(N) primes up to N. Although this was close, it was found to gradually drift away from the true number of primes as N got larger and larger. Here is a comparison of Gauss’s childhood guess, shown as the lower plot, with the true number of primes, the upper plot:

      A comparison between Gauss’s guess and the true number of primes.

      This graph reveals that although Gauss was on to something, there still seemed to be room for improvement.

      Legendre’s improvement was to replace the approximation N/log(N) by

      thus introducing a small correction which had the effect of shifting Gauss’s curve up towards the true number of primes. As far as the values of these functions that were then within computational reach, it was impossible to distinguish the two graphs of π(N) and Legendre’s estimate. Legendre, steeped in the prevailing preoccupation with the practical application of mathematics, was much less reluctant to stick his neck out and make some prediction about the connection between primes and logarithms. He was not a man scared to circulate unproven ideas, or even proofs with gaps in them. In 1808 he published his guess at the number of primes in a book about number theory entitled Théorie des Nombres.

      The controversy over who first discovered the connection between primes and logarithms led to a bitter dispute between Legendre and Gauss. It was not limited to the argument about primes – Legendre even claimed that he had been the first to discover Gauss’s method for establishing the motion of Ceres. Time and again, Legendre’s assertion that he had uncovered some mathematical truth would be countered by an announcement by Gauss that he had already plundered that particular treasure. Gauss commented in a letter of July 30, 1806, written to an astronomical colleague named Schumacher, ‘It seems to be my fate to concur in nearly all my theoretical works with Legendre.’

      In his lifetime, Gauss was too proud to get involved in open battles of priority. When Gauss’s papers and correspondence were examined after his death, it became clear that due credit invariably went to Gauss. It wasn’t until 1849 that the world learnt that Gauss had beaten Legendre to the connection between primes and logarithms, which Gauss disclosed in a letter to a fellow mathematician and astronomer, Johann Encke, written on Christmas Eve of that year.

      Given the data available at the start of the nineteenth century, Legendre’s function was much better than Gauss’s formula as an approximation to the number of primes up to some number N. But the appearance of the rather ugly correction term 1.083 66 made mathematicians believe that something better and more natural must exist to capture the behaviour of the prime numbers.

      Such ugly numbers may be commonplace in other sciences, but it is remarkable how often the mathematical world favours the most aesthetic possible construction. As we shall see, Riemann’s Hypothesis can be interpreted as an example of a general philosophy among mathematicians that, given a choice between an ugly world and an aesthetic one. Nature always chooses the latter. It is a constant source of amazement for most mathematicians that mathematics should be like this, and explains why they so often get wound up about the beauty of their subject.

      It is perhaps not surprising that in later life Gauss further refined his guess and arrived at an even more accurate function, one which was also much more beautiful. In the same Christmas Eve letter that Gauss wrote to Encke, he explains how he subsequently discovered how to go one better than Legendre’s improvement. What Gauss did was to go back to his very first investigations as a child. He had calculated that amongst the first 100 numbers, 1 in 4 are prime. When he considered the first 1,000 numbers the chance that a number is prime went down to 1 in 6. Gauss realised that the higher you count, the smaller the chance that a number will be prime.

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