The Music of the Primes: Why an Unsolved Problem in Mathematics Matters

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When reading Maths books, I think it’s important to distinguish between books about Maths or the history of Maths, and books where you actually get to learn and do Maths. You should try to have a good balance of both types on your personal statement. As you can tell The Music of the Primes is one of the books about the history of Maths. Prime numbers become less frequent as numbers get larger. There are fewer in any interval greater than let’s say 1000, than the same interval less than 1000. This is intuitively obvious since the greater the number the more lesser numbers there that might be divided into it evenly. Interestingly, there is always at least one prime between any number and its double.

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

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. Gauss's guess is like the prediction that a six-sided dice thrown 6,000 times lands exactly 1,000 times on the prime side. The heights of Riemann's harmonic waves tell us how far Gauss's guess is from the way the prime number dice really landed, that is, the errors between Gauss's guess and the true number of primes. 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) Riemann was very shy as a schoolchild and preferred to hide in his headmaster's library reading maths books rather than playing outside with his classmates. It was while reading one of these books that Riemann first learnt about Gauss's guess for the number of primes one should encounter as one counts higher and higher. Based on the idea of the prime number dice, Gauss had produced a function,

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 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.

The music of the primes | plus.maths.org

This book was at its heart a biography of the Reimann Hypothesis, and of the mathematicians who worked on trying to prove or disprove it over the years. I really liked the way that it showed the relationships among the people involved, and how the centers of number theory research shifted from Paris to Göttingen to Princeton, and how this was caused in large part by the geopolitics of the area (Napoleon and Hitler in particular). the book explores The Riemann Hypothesis which is mainly a problem of navigating the primes looking for a pattern.

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Una cosa que no me ha gustado es el abuso que hace a veces el autor de la analogía. Es difícil divulgar sobre matemáticas, y más sobre matemáticas complejas como la teoría de números. Hay que encontrar un equilibrio entre lo demasiado simple y lo demasiado farragoso. Pero al autor, a veces, se va no ya por lo simple sino por lo incomprensible. Cuando habla de la intersección no nula de los números primos y la física cuántica, hace una analogía con "una tambor cuántico", que no queda del todo clara. Pero a partir de ese momento sólo hablará de físicos y matemáticos diversos que investigan sobre tambores cuánticos, así sin comillas. ¿Tambores cuánticos? ¿No podría el autor definir algo más en serio, aunque fuera una vez, a qué se refiere exactamente con un tambor cuántico, y luego ya seguir con la analogía? Otra de estas analogías son las "calculadoras de reloj", que usa sin comillas a lo largo de todo el libro para referirse a la aritmética modular. Como en un reloj de 12 horas 9+4 o es 13 sino 1 (y así nos introduce la aritmética modular), cualquier referencia posterior a la aritmética modular la traviste de calculadoras de reloj. Son dos analogías sobreutilizadas que recuerdo que no me gustaron. En cualquier caso, nadie ha dicho que sea fácil divulgar ideas tan complejas. Su punto de de equilibrio entre lo preciso y lo comprensible para el público está un poco más escorado que el mío. urn:lcp:musicofprimessea00dusa:epub:5fb70ea0-81ab-439c-af77-d932c2cc22dc Extramarc Notre Dame Catalog Foldoutcount 0 Identifier musicofprimessea00dusa Identifier-ark ark:/13960/t26b04263 Invoice 11 Isbn 9780066210704

music of the primes - maths The music of the primes - maths

If there is advanced technological life elsewhere in the universe, it would unlikely be Christian, or Muslim, or Jewish, or Buddhist. It would however certainly know the same mathematics that we do. And it would understand the phenomenon of the prime numbers and their significance as much as, perhaps more than, we do. Mathematics is the natural religion of the cosmos; and prime numbers are its central mystery. Riemann had found one very special imaginary landscape, generated by something called the zeta function, which he discovered held the secret to prime numbers. In particular, the points at sea-level in the landscape could be used to produce these special harmonic waves which changed Gauss's graph into the genuine staircase of the primes. Riemann used the coordinates of each point at But where on earth had Riemann found these strange prime number harmonics which corrected Gauss's guess into the true sound of the primes? Well, he was actually messing about with an exciting new subject that was emerging out of the French Revolution: the new world of imaginary numbers. For years people could not accept that a negative number might have a square root - after all, a In 1859 Bernard Reimann published his hypothesis on prime numbers; that the real part of any non trivial zero of the Riemann zeta function is 1/2. It was apparently proven by Riemann himself but the proof was never found, reportedly burned by his housekeeper when tidying up after his death. Since then many mathematicians have devoted their efforts to providing enough evidence that this is true. Even with the advent of supercomputers and the finding of prime numbers with a million digits, which still fulfil the hypothesis, it has not been proven satisfactorily. Attempts to disprove it have been equally unsuccessful by not finding a single prime number that doesn't behave in this way.

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Mathematicians feel like characters and the course of history feels like a fictional story beautifully woven by du Sautoy. Although the book does not delve into any theory, it is tough not to keep reading about each of the protagonists and their achievements on the side. It is tough to get out of the loop. Wikipedia, Numberphile, 3Blue1Brown are some of the resources that I would suggest to go along with the book. However, I felt more and more at sea as the book went on. Given that I have studied the Riemann Hypothesis at Masters level, and even written an essay on it and the Riemann Zeta Function (in 2019), you would think I’d do better – however, my maths brain has not done well since I gave up in 2021, and I have forgotten so much. 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. Many people have commented over the ages on the similarities between mathematics and music. Leibniz once said that "music is the pleasure the human mind experiences from counting without being aware that it is counting". But the similarity is more than mere numerical. The aesthetics of a musical composition have much in common with the best pieces of mathematics, where themes are