At one stage, young Lemaître was being supervised by the famous number theorist de Vallée-Poussin, who appears in this book. Impressed by his precocious talent, de Vallée-Poussin suggested to Lemaître that he should try to solve the Riemann conjecture as his thesis project. But the Riemann conjecture was just too hard. _______________________________ Inspired by yesterday's discussion with Nandakishore and Matt and the famous Up Goer Five XKCD cartoon (soon to become a book), I wrote the following:Can I explain Mr Riemann's very important number idea with only the ten hundred most used words? _______________________________ Following on from that, is it possible to explain the Riemann Hypothesis to an eight-year-old? The conversation went roughly as follows: - So Jenkyn, do you know what a prime number is? You can turn some numbers into rectangles. - Well, those are called prime numbers. What else do you think would be a prime number?
Remember the math major in your college dorm, the one doing advanced physics with more Greek symbols than Roman numerals? Well, the book Prime Obsession deals with mathematical concepts magnitudes of order more complex than those brainiacs could ever wish to comprehend. When I saw several books about the RH, I read a dustcover and, at once, needed to be educated about a theorem that's remained relatively unchanged, and is still only as advanced as when it was presented by Riemann 150 years ago, despite Cray-1 supercomputers crunching possible solutions for 40 years and programmed by myriad math geniuses produced by the 20th century. Either way, I figured the book would give me a few buzz words to finesse around a cocktail party in the extraordinary event math was discussed over finger foods and wine. --the Euler-Mascheroni number --the Rieman Hypothesis: All non-trivial zeros of the zeta funtion have real part one-half --the Golden Key --the sieve of Eratosthenes --the Prime Number Theorem, pi(N)Li(N) --complex conjugates --zeta function critical strip --Gram's zeros --Riemann surface --value plane from the critical line --Big Oh and Mobius Mu --Matricies (eigenvalues, trace, characteristic polynomials of) --operators --Guassian-random Hermitan matricies --Guassian Unitary Ensemble --Chaos theory --And many, many, many complex, irrational formulas without Roman numerals To give Derbyshire credit, what Prime Obsession attempts to do is chronologize the different attempts by which mathematicians have tried to solve the RH.
Basically, in his analytically continued zeta-function, taking in complex values, he says that all the non trivial zeros have a real part one-half. More interesting results for me was the expression of zeta in terms of a product solely involving an expression of primes, and the analytic continuation of the zeta function, which you can generate by taking the more-likely-to-converge eta function and manipulating a bit, or taking this, which I have no idea how to derive from anything: Also the gamma function is mind-blowing.... So apart from the math that I still don't really understand, I got a pretty good survey of the interesting historical figures that have populated math since Euler, including Gauss, Dirichlet, Riemann himself, Dedekind, Hilbert, (these five excluding Euler surrounded by the quirky atmosphere of Gottingen which I imagined as a black gothic fort), i forgot what Poussin Hadamard Chebyshev and Landau did, and then of course you have math's Tom and Jerry - Hardy and littlewood, who proved that there were a infinite number of non-trivial zeroes and that Li(x) intersects with pi(n) an infinite number of times (which was very surprising), respectively. :( (Apparently this is a way of saying that if the zeta zeroes don't misbehave, the error term does not misbehave, which was not explicit in Riemann's formulations) 2. 4. And fina-fuckin-lly we return to Riemann who manages to formulate J(x) in terms of the zeta function (I thought we already did that????)(and how exactly do you do that????? So now we have a way of formulating pi(x) in terms of J(x) in terms of zeta (by means of the non-trivial zeta zeroes, in terms of sum of xall the roots). So we have this really mysterious relation between the values of the zeroes and the more significant "error term" of J(x).
You need not to know much mathematics to start reading it, he teaches you along the way.
I guess the truth is that I would rather have just had a really good story than have to also hear what Derbyshire thought about it himself. Which is not to say there isn't a good story here!
Even the mathematics of Riemann from the middle of the 19th c.