music of the primes

Let me admit it : i was probably wrong in this post to advise against downloading A walk in the noncommutative garden by Alain Connes and Matilde Marcolli. After all, it seems that Alain&Matilde are on the verge of proving the biggest open problem in mathematics, the Riemann hypothesis using noncommutative geometry. At least, this is the impression one gets from reading through The music of the primes, why an unsolved problem in mathematics matters by Oxford mathematician Prof. Marcus du Sautoy… At the moment I’ve only read the first chapter (Who wants to be a millionaire?) and the final two chapters (From orderly zeros to quantum chaos and The missing piece of the jigsaw) as I assume I’ll be familiar with most of the material in between (and also, I’m saving these chapters for some vacation reading). From what I’ve read, I agree most with the final review at amazon.co.uk

Fascinating and infuriating, October 5, 2004
Reviewer: pja_jennings from Southampton, Hants. United Kingdom
This is a book I found fascinating and infuriating in turns. It is an excellent layman’s history of number theory with particular reference to prime numbers and the Riemann zeta function. As such it is well worth the reading.
However I found that there are certain elements, more of style than anything else, that annoyed me. Most of the results are handed to us without any proof whatsoever. All right, some of these proofs would be obviously well beyond the layman, but one is described as being understandable by the ancient Greeks (who started the whole thing) so why not include it as a footnote or appendix?
Having established fairly early on that the points where a mathematical function “reaches sea level” are known as zeros, why keep reverting to the sea level analogy? And although the underlying theme throughout the book is the apparent inextricable link between the zeta function’s zeros and counting primes, the Riemann hypothesis, I could find no clear, concise statement of exactly what Riemann said.
Spanning over 2000 years, from the ancient Greeks to the 21st century, this is a book I would thoroughly recommend.

Books on Fermat’s last theorem (and there are some nice ones, such as Alf Van der Poorten’s Notes on Fermat’s last theorem) can take Wiles’ solution as their focal point. Failing a solution, du Sautoy constructs his book around an April’s Fool email-message by Bombieri in which he claimed that a young physicist did prove the Riemann hypothesis after hearing a talk by Alain Connes. Here’s du Sautoy’s account (on page 3)

According to his email, Bombieri has been beaten to his prize. ‘There are fantastic developments to Alain Connes’s lecture at IAS last wednesday.’ Bombieri began. Several years previously, the mathematical world had been set alight by the news that Alain Connes had turned his attention to trying to crack the Riemann Hypothesis. Connes is one of the revolutionaries of the subject, a benign Robespierre of mathematics to Bombieri’s Louis XVI. He is an extraordinary charismatic figure whose fiery style is far from the image of the staid, awkward mathematician. He has the drive of a fanatic convinced of his world-view, and his lectures are mesmerising. Amongst his followers he has almost cult status. They will happily join him on the mathematical barricades to defend their hero against any counter-offensive mounted from the ancien regime’s entrenched positions.

Contrary to physics, mathematics doesn’t produce many books aimed at a larger public. To a large extend this is caused by most mathematicians’ unwillingness to sacrifice precision and technical detail. Hence, most of us would never be able to come up with something like du Sautoy’s description of Weil’s work on the zeta function of curves over finite fields (page 295)

It was while exploring some of these related landscapes that Weil discovered a method that would explain why points at sea level in them like to be in a straight line. The landscapes where Weil was successful did not have to do with prime numbers, but held the key to counting how many solutions an equation such as $y^2=x^3-x$ will have if you are working on one of Gauss’s clock calculators.

But, it is far too easy to criticize people who do want to make the effort. Books such as this one will bring more young people to mathematics than any high-publicity-technical-paper. To me, the chapter on quantum chaos was an eye-opener as I hadn’t heard too much about all of this before. Besides, du Sautoy accompanies this book with an interesting website musicofprimes and several of his articles for newspapers available from his homepage are a good read (in case you wonder why the book-cover is full of joggers with a prime number on their T-shirt, you might have a look at Beckham in his prime number). The music of the primes will definitely bring many students to noncommutative geometry and its possible use to proving the Riemann Hypothesis.

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