The Music of the Primes, by duSautoy.
> Here is something related to all that, the Riemann Zeta function. I
> don't really understand it, but it is fascinating. Go here:
>
> http://en.wikipedia.org/wiki/Reimann_hypothesis
>
Funny you should bring that up... it's the reason I am reading this book, (among others I have read in the past), since a famous visiting professor once slipped the Riemann Hypothesis on us as a problem we might wish to attempt solving. I had no clue on how to even get going, but some of the brighter and more dedicated fellows spent hours poring over it, until a professor in the department let them on to what this beast was!
The Zeta function is an infinite summation/series: 1 + 1/2^n + 1/3^n + ... applied to the complex numbers. The interesting thing is that Euler broke this down into a prime product (since any number, in this case the denominators, is the product of primes). You can see that for real numbers, for n = 1 this is a divergent series (whose sum is infinity) and that for n > 1, it is convergent (values for n=2, n=3, etc have been discovered). Reimann hypothesized that the non-trivial values of n for which this sum evaluates to zero are ones in which the real part "x" of the complex number (x+iy) is 1/2. You find these zeros as you iterate over (well range over) the real numbers for 'y'. These zeroes turn out to be valuable in terms of their relationship to the prime counting function. The prime counting function counts the number of primes that exist below any given number 'n'. And of course things to do with prime numbers are the Holy Grail of mathematics ;-).
Hilbert posed the Riemann Hypothesis as one of the most important problems for mathematics in the 20th century, in his famous lecture in 1900 to the world mathematical congress. He is supposed to have said that if he were to fall asleep and wake up 500 years later, his first question would be whether a solution had been found for the Riemann Hypothesis.
> If you're at all interested in this...? I am writing from the repair
> shop, so this is a little sketchy.
I am looking over the links you provided. They are interesting, though a bit over my head. And I am afraid I was not quite able to follow your notes on them (snipped out above) well either. My mathematics knowledge is fading fast...
--ravi