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!... ravi
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Well, I knew it had to be something like that. It's just one of the most famous and most difficult problems around. And primes are full of these kinds of problems: deep, hard, fundamental, and very very difficult to even understand what the question is.
Any way the very unclear points to the quick post I did at work today were several. First of all, let me be clear. I absolutely do not understand this problem. But I think I have a vague mystical sense of it (that sounds silly, doesn't it?). Second, whenever I've seen problems that are utterly opaque, I always look for a different representation, a representation in a form that I feel more at home with. And lastly, in a different respresentation, I next try to figure out what makes the two representations transform from one to other and back. That collection of invariants (or symmetries in a different language), is often a clue on how to understand the problem.
When Wiles solved Fermat's Last Theorem, he and others approached the problem in a similar way to the one I outlined above, but of course on a vastly more sophisticated and knowledgable way. They didn't start in the number theory. They started in the algrebraic geometry of elliptic curves. I think the reasoning was, if they could show such and such was true for elliptic curves (a different and related representation), then they could tranform that proof into one that applied to Fermat's Last Theorem. (With many sideline proofs in yet other branchs along the way---none of which I could follow.)
So, then I thought, maybe that's the way to understand Riemann's Zeta function. Here's the very crude reasoning. Anything that has the form z = 1/z can be represented as an inversion, a reciprocal, a conformal transformation, and will involve circles, conic sections, polars, points at infinity, projective spaces, and so forth. Since none of these sorts of geometric ideas require the archane nonesense of imaginary numbers they are in some sense simplified (at least for me). That what the links on polars were about.
Let's go back to the harmonic series. The most interesting thing about the harmonic series is that it doesn't converge. But the reason is even more interesting. It doesn't converge because zero, where the limit should intuitatively be, isn't in the fucking interval. The way to see this is to use the alternating harmonic series, where both positive and negative 1/n's alternate. The alternating harmonic does converge to zero. Why? Because if you think about it, zero is not only in the interval, but it is approached from both sides of the number line. It can not escape being sandwiched in between the negative 1/n's and the positive 1/n's. In other words, the reason the harmonic does not converge is that while it may have a greatest lower bound (1/inf), the least upper bound (0) doesn't exist and therefore is not on the interval, so it can not form a limit. The alternating harmonic does not have this problem, because both bounds are present on the interval. In other words the interval (by hidden assumption) doesn't contain 0, so how could the series converge to it? BTW, I think this illustrates a fundamental tautology in the theory of limits, i.e. the limit must be assumed to exist on the interval, prior to its demonstration.
Most of the time my mind works almost exclusively with geometry. I think almost entirely in spatial terms. Luckily, many things in number theory have geometric representations. In the above case of the harmonic series, while I followed the elementary calculus version, (using the ratio test for convergence), what I really did was construct a figure that generated the harmonic series as line divisions: 1/1, 1/2, 1/3,...1/n, then reflected it onto the negatives -1/1, -1/2...-1/n, so I could see what was happening. It's a very crude way to think, but it makes the problem much more obvious to me.
Anyway, I think the basic interest in the primes is this. Because all numbers not prime, can be factored into primes, it is simply inconceivable that the natural occurance of primes is not itself based on some kind of module.
``My mathematics knowledge is fading fast...'' Well, it's problems like this that get it back and add to it. Of course I am no one to talk. I got into a whole trip on ancient math, just because I couldn't figure out how to build a set of stairs between a high deck and the backyard.
You know, the Courant and Robbin's book, What is Mathematics is pretty good, and has a lot of interesting pieces of number theory, geometry and analysis in it.
CG