The fun arises because although mathematicians know primes occur less and less frequently as we progress up the scale of numbers, no one knows how to predict when the next one will be encountered. They can be, and have been, calculated to very large numbers indeed, but they can’t be anticipated, only recognised once they appear.* Or should the term be ‘revealed’?

The Music of the Primes - Wikipedia

training can respond to a concert performance, whereas only after years of mathematical training does one eventually have the ears to listen to the great mathematical compositions. Riemann's early death deprived mathematics of one of the giants of its subject. Just as the world was denied the music of a mature Schubert who died at the same age as Riemann, the world is still waiting for a successor to capitalise on the insights generated by Riemann in his attempts to capture the music of the primes. We take it for granted now that evolutionary biology, among other things, helps us understand human behaviour, but we're not entirely sure why maths matters - if, indeed, it matters at all. Hence books like this, which strain to assert their importance: 'Why an Unsolved Problem in Mathematics Matters'. Hence Marcus du Sautoy, whose combination of brains and charm should soften up even the most wilfully innumerate of readers.So far, so good. I am not a mathematician and, even now, I could not explain to you the derivation and use of a zeta function - there may be none for all I know. This is a book as much about mathematicians as it is about their subject matter, and they are every bit as fascinating. These are people who are so driven by the abstract that they seem to want to find the rules that govern even the most random events using a language that has evolved in huge leaps to the point of being unrecognisable by ordinary men.

The Music of the Primes: Why an Unsolved Problem in

Prime numbers and their distribution have always been one of the more interesting subjects to talk about. This book takes you through the whole journey of starting out with finding the first few prime numbers to trying to find a pattern on how primes are spread through the universe of natural numbers. The list of protagonists include Euclid, Euler, Gauss, Riemann, Polignac, Hilbert, Hardy, Littlewood, Ramanujan, Godel, Turing to name a few. Naturally, the book focuses on one of the most important conjectures ever : The Riemann Hypothesis. He sets himself quite a task, though. The Music of the Primes is about the search for a formula which will enable mathematicians to understand the distribution of prime numbers. Primes, you will remember, are those numbers divisible only by one and themselves - 2, 3, 5, 7, 11, 13, 17, 19, etc... - although it's not as simple as that 'etcetera' might suggest. While other number sequences continue in predictable ways, primes can still only be located through a laborious process of trial and error. There is no formula for finding the six billionth prime, for instance, although a computer, going through all the other numbers on the way, will get there eventually. The highest prime yet discovered is a number with more than four million digits. One of the great symphonic works of mathematics is the Riemann Hypothesis - humankind's attempt to understand the mysteries of the primes. Each generation has brought its own cultural influences to bear on its understanding of the primes. The themes twist and modulate as we try to master these wild numbers. But this is an unfinished symphony. We still await the mathematician who can add the final frequencies. This time the sine waves must fit the length of the clarinet but be open at one end, closed at the other. This results in the clarinet choosing a different sequence of harmonic notes to those favoured by the violin. I think it’s really easy as a young mathematician to decide you only like one side of maths and neglect the other but school maths does not give you enough to go on. I think it’s important to keep an open mind and this book helped me appreciate applied maths when I read it in Year 12. Before then I immaturely decided I would specialise in pure maths without really considering what applied maths would be like at university.

Gowers, W. T. (October 2003), "Prime time for mathematics (review of Prime Obsession and The Music of the Primes)", Nature, 425 (6958): 562, doi: 10.1038/425562a

Music of the Primes by Marcus du Sautoy | Perlego [PDF] The Music of the Primes by Marcus du Sautoy | Perlego

A YouTube video I found very useful for visualising the Riemann Zeta Function (it is really stunning, and well worth a look) is by 3Blue1Brown: “Visualising the Riemann zeta function and analytic continuation” https: // www.youtube. com/watch? v=sD0NjbwqlYw (remove the spaces) But Riemann couldn't prove that every point at sea level really lay on this magic leyline (or "critical line", as mathematicians call it) that seemed to be running through his landscape. But he hypothesised they did. And this is what all mathematicians would sell their souls to prove - even without the million dollar prize that has been offered for a solution. The Riemann Hypothesis: Prime numbers are the very atoms of arithmetic. They also embody one of the most tantalising enigmas in the pursuit of human knowledge. How can one predict when the next prime number will occur? Is there a formula which could generate primes? These apparently simple questions have confounded mathematicians ever since the Ancient Greeks. Desde Euclides, que demostró que los números primos son infinitos (hoy el más elevado es 2 elevado a 13.466.917 - 1, hallado en 2001 por un estudiante canadiense, un número de cuatro millones de cifras), hasta Euler en San Petersburgo, el trío de Gotinga (Gauss, Riemann, Dirichlet), Cauchy, las series armónicas de Pitágoras, Fourier, Hilbert, Hardy, Skewes, Ramanujan (el matemático Indio de Cambridge, que fue protagonista de una película reciente), Gödel y su teorema de la incompletitud, las máquinas de Touring, la criptografía RSA y la relación entre los primos y la física cuántica. Un recorrido inacabado y muy bien contado.

Again by adding the heights of all these sine waves together we can see the square shape of the clarinet emerging from the basic sine wave corresponding to the A of the tuning fork. Follow this link to see the way the first five harmonics combine to build up the wave shape created by a clarinet. negative times a negative is always positive. But the French revolution gave mathematicians the courage to think of new ideas. They invented new months and new days of the week, so why not new numbers? So came about the birth of the new number i, the square root of minus one. All the other imaginary numbers were got by taking combinations of this new number with the ordinary numbers, for