The German Genius

fascination with prime numbers was introduced, and it was noted how Gauss had uncovered the link between primes and logarithms. His invention of imaginary numbers was also discussed (see Chapter 7). It will be remembered that the link between primes and logarithms allowed Gauss to give a good but still approximate prediction of how many primes there are up to any figure, N . It was Riemann’s achievement to establish and clarify a definitive prediction of the number of primes—and by using Gauss’s other invention, imaginary numbers.
    Riemann worked with something known as the zeta function. This, in one form or another, had been of interest to mathematicians since Pythagoras, in ancient Greece, who had pointed out the link between mathematics and music. Pythagoras found that if he filled an urn with water and banged it with a hammer, it would produce a certain note. If he then removed half the water and banged the urn again, the note had gone up an octave. As he removed water equivalent to a proper integer ( 1 /2, 1 /3, 1 /4), the notes produced sounded in harmony to his ear, whereas if any intermediate amount were removed, the sound was discordant. Pythagoras came to believe that numbers lay at the root of the order of the universe, giving rise to his famous phrase “the music of the spheres.”
    For other mathematicians, however, this led to an investigation of the behavior of reciprocal numbers (the reciprocal of 2 is 1 /2 and the reciprocal of 3 is 1 /3). This investigation eventually led to what mathematicians call the zeta function, zeta being represented by the Greek symbol,. The zeta function is represented as follows:

     
    This function turns up some interesting results, the most celebrated being the discovery in the eighteenth century by the Swiss mathematician Leonhard Euler that when zeta is 2 the sequence becomes:

     
    and that this may eventually be written as:

     
    This discovery took the mathematics world by storm, for the number, 1 / 6 2 , written as a decimal, produces an indefinite progression, likeitself. (This is number theory, remember, the sheer behavior of numbers being fascinating for mathematicians, whether that behavior has any use or not.) 37
    When Riemann was elected to the Academy in Berlin in November 1859, he wrote a ten-page paper to mark the event, as was (again) customary. This proved every bit as radical as his inaugural lecture. One of the things he did in this paper was to feed Gauss’s other invention—imaginary numbers—into the zeta function, obtaining an entirely unexpected pattern, the most notable feature of which was (when the results of the equations were plotted on a graph) a series of waves and which, he found, could be used to correct Gauss’s calculations regarding primes, to give an exact , error-free prediction of the number of primes in any sequence. And so the apparent randomness of the primes had been shown to have an order. Not a simple order, it is true, but an order nonetheless. Order—however complex—is a form of beauty for mathematicians.
    Felix Christian Klein was born in Düsseldorf on April 25, 1849, and delighted in pointing out that his birth date was a collection of primes squared: 5 2 , 2 2 , 7 2 . He made his most important contribution in group theory, another new field. The son of a government official in the Rhine province, he was appointed to a professorship, in his case at Erlangen, at an even younger age than Boltzmann, when he was twenty-three. He moved to Munich’s Technische Hochschule in 1875, where he taught, among others, Max Planck (he also married Anne Hegel, granddaughter of the philosopher). In 1886 his health deteriorated, and he accepted a quieter life as professor of mathematics at Göttingen. There he consolidated Göttingen as the world’s leading mathematics research center. 38
    To explain what Klein was driving at in group theory, imagine two visual experiments. First, imagine a rectangular sheet of paper, its sides measuring A and B inches. Rotate the sheet through forty-five degrees and then photograph it. The photograph will not show a rectangle and the sides will not be A and B inches long, yet the paper will not have changed. What are the mathematics of this foreshortening, the relationship between the original and the photograph? Second, consider an aerial photograph of a particular country—for example, Italy—taken from a satellite fifty miles up in space. Next, view the same country from, say, five