Why an Unsolved Problem in Mathematics Matters
In 1900, David Hilbert presented a challenge to mathematicians around the world: 23 unsolved problems to defeat by the end of the twentieth century. Since then, most of these problems have crumbled, but one remains stubbornly and famously unsolved. This problem is the Riemann hypothesis, the focus of Marcus Du Sautoy’s elegant book, The Music of the Primes.
Prime numbers (numbers only divisible by one and themselves) are the building blocks of all natural numbers. Humans have been interested in primes for millennia; Euclid proved that there are infinitely many and Eratosthenes devised a simple method for ‘sieving’ them out. Despite our recognition of the fundamental importance of primes, they have always been unpredictable. Where does the next one appear?
The Riemann hypothesis, proposed in 1859, predicts how many primes exist up to a particular number. This involves summing over ‘zeroes’ (solutions) of the Riemann zeta function. Bernhard Riemann found that the zeroes stood precisely on a line which stretched as far as mathematics would allow us to see. He proposed that the position of prime numbers were controlled by the hiding places of these zeroes. Order seemed to arise from the chaos of the primes, or to use Du Sautoy’s musical analogy, the Riemann hypothesis makes harmony from a cacophony.
So far, we know that the first 10,000,000,000 zeroes of the Riemann zeta function stand in line. But this is not proof enough for a mathematician, and the bulk of this book documents the attempts of various geniuses to uncover a proof. We meet fascinating twentieth century mathematicians who took on the Riemann hypothesis: there is the charming G.H. Hardy with his personal vendetta against God; the academic hobo Paul Erdõs who wrote papers with 500 different people; the Laurel and Hardy-esque Birch and Swinnerton-Dyer; the sickly genius Srinivasa Ramanujan, who taught himself mathematics in poverty and isolation; and the equally tragic Alan Turing, who attempted to use the first stored-programme digital computer to disprove the Riemann hypothesis. Du Sautoy describes these individuals with erudition and affection. By doing so, he transforms these mathematicians from names attached to hypothesis into funny, frustrating and awe-inspiring characters.
By leading us on a journey through the lives of mathematicians, The Music of the Primes is more than just the story of the Riemann hypothesis; it is the story of modern mathematics. We learn how mathematics fared under Napoleon, the spirit of Cambridge in the early twentieth century, how the dark city of Gottingen was lit by the greatest mathematical minds in the world and how this was brutally snuffed out by the emergence of the Nazi Party. The elaborate scene-setting could become frustrating for readers who just want to learn the mathematics behind the Riemann hypothesis.
The Music of the Primes is largely about the elegance of patterns found in mathematics, but for those who remain unconvinced, Du Sautoy explains how prime numbers are useful. Online banking and e-commerce are made possible through the use of staggeringly large primes and the ‘Gaussian clock’. Du Sautoy also deviates from pure mathematics to describe the exciting appearance of the primes in nature: not only have animals and plants evolved to use them, but the primes seem to appear in the fundamental fabric of the universe: the spacing between prime numbers may correspond to the spacing of energy levels of atoms.
The beauty of the language in The Music of the Primes is on par with Dawkins’ in Unweaving the Rainbow. Complicated mathematics is compared to music; numbers are notes and the Riemann hypothesis transforms chaotic noise into a symphony (the music of the primes). Du Sautoy continues to use the language of music through the book, creating a rich analogy which gives the Riemann hypothesis aesthetic appeal usually only appreciated by mathematicians. The description of the Riemann zeta function as a ‘landscape’ is similarly poetic; Du Sautoy describes a mysterious ‘looking glass world’ with mountains and valleys, in which mathematicians have discovered zeros sitting at sea level in line stretching from North to South. Many readers, however, will find themselves lost in the metaphors and forgetting the mathematical meaning of terms such as ‘sea level’ or ‘landscape’ halfway through the book.
When you read The Music of the Primes, do not expect a rigorous mathematical examination of the Riemann hypothesis; if it is rigour you want, your time would be better spent on academic publications. However, you can expect to learn about the importance of prime numbers and the Riemann hypothesis, the story of great minds and how they failed to defeat it and why prime numbers are not just essential but elegant.