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The Music of the Primes: Searching to Solve the Greatest Mystery in Mathematics ReviewOne of the attractions of number theory is that it has to do with the counting numbers; if you can get from one to two and then to three, you are well on your way to hitting all the subject matter of "The Queen of Mathematics." All those numbers can be grouped into two simple categories. The composite numbers, like 15, are formed by multiplying other numbers together, like 3 and 5. The prime numbers are the ones like 17 that cannot be formed by multiplying, except by themselves and 1. Those prime numbers have held a particular fascination for mathematicians; they are the atoms from which the composites are made, but they have basic characteristics that no one yet has fully fathomed. We know a lot about prime numbers, because mathematicians have puzzled over them for centuries. We know that as you count higher and higher, the number of primes thin out, but Euclid had a beautiful proof that there is no largest prime. However, the primes seem to show up irregularly, without pattern. Can we tell how many primes are present below 1,000,000 for instance, without counting every one? How about even higher limits? Speculating about the flow of primes led eventually to the Riemann Hypothesis, the subject of _The Music of the Primes: Searching to Solve the Greatest Mystery in Mathematics_ (HarperCollins) by mathematician Marcus du Sautoy. The counting numbers turn out to be astonishingly complicated, and Du Sautoy knows that egghead number theorists will understand these complications better than we nonmathematicians, but he invites us to consider at a layman's level the importance of the particular quest of proving the Riemann Hypothesis. He is convincing in his demonstration that it is worth knowing what all the effort is about.Bernhard Riemann, a mathematician at the University of Gottingen, introduced a "zeta function," and proposed that when this particular function equals zero, all the zeros will wind up on a specific line when graphed on the complex plane. Further effort has shown that there are millions of zero points on that line, just as the hypothesis says, and no zero points have been found off the line. Neither of these facts makes a proof, however. Du Sautoy wisely shows some of the enormously complex technicalities of the speculations and computations, but makes no attempts to try to get the reader to comprehend the hypothesis at the level he does. There are a number of reasons that the proof is so important. Right now there are a large number of tentative proofs of important mathematical ideas; they are all based on the Riemann Hypothesis being true, but of course, it has not itself been proved. A proof would tell us more about the prime distribution and finding primes, and this subject has become vital since cryptography, including how you privately send your credit card number across the internet, is based on prime numbers and the difficulty of factoring two big primes multiplied together. The way the Riemann zeros are distributed seems to mirror the patterns quantum physicists find among the energy levels of the nuclei of heavy atoms; in proving Riemann, we may have a closer understanding of fundamental reality.
With the Riemann Hypothesis central to a lot of mathematical effort, Du Sautoy is able to bring in a lot of side issues, such as Turing's attempt to find a program that would attack the proof, the four color map theorem and computer proofs in general, Gödel's Incompleteness Theorem, and much more. The mathematics, such as it is, is leavened by portraits of mathematicians, who range from conventional to very peculiar. A good deal is said about the dashing Italian mathematician Enrico Bombieri who rocked the mathematical world with the announcement that the Riemann Hypothesis had finally been proved. There was jubilation over the announcement until mathematicians realized that the e-mail bore the date 1 April. He could not have picked a better theme for an April Fool's joke; all the mathematicians were eager to see this one proof finally nailed down. Readers who take du Sautoy's entertaining tour can get an idea of why all the effort is being expended on the proof, and what elation there will be if it is ever found.The Music of the Primes: Searching to Solve the Greatest Mystery in Mathematics Overview
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