I just watched a show that expresses a lot of excitement about the Riemann Zeta Function, and I wondering why care any more about the mysteries of the Riemann Zeta Function the same mystery of Fibonacci Sequence? Both demonstrate numbers can represent physical living reality. The Riemann Zeta Function helps us understand atoms, but the Fibonacci sequence helps us understand living things such as the growth of corn, so why care more about than the other?
It is somewhat a matter of personal preference but in terms of which turns up more in things like physics or has applications to things like security the Zeta function wins by a mile. The golden ratio and the Fibonacci sequence arises in a very simple manner and we see things pertaining to it in a few living entities but it doesn't have the 'deep' structure the Zeta function has. The Zeta function relates to primes via the Euler formula and due to a few more complicated results it is reasonable to expect a proof to the Riemann hypothesis, which is about the Zeta function, will provide significant insight into the property of primes. Internet security is based on certain properties of primes so if you can find prime factors quickly you can basically invalidate SSL encryption. Amazon and your bank might have issues then... The function also arises when you want to consider certain properties of space-time and quantum field theories. The reason string theory says space-time is 10 dimension is a direct and immediate consequence of the fact the singularity at z=-1 of the Zeta function has residue -1/12. If string theory is right then the properties of space-time are intimately linked to the Zeta function. I'm assured by a number of functional analysis and number theory PhDs I know that there's lots of interesting mathematics and deep generalisations arising from the Zeta function in those domains but I lapse into a coma when they start talking about Dirichlet L functions and Hecke operators for modular forms. I struggle to think of the last time I came across the golden ratio in a physics area, it seems to become less interesting as you learn more maths. The basic concept is pretty simple and hence why it has such a prominent place in the minds of laypersons or worse, internet hacks who think they've got a theory of everything because they've discovered something amazing about \(\frac{1}{2}(1+\sqrt{5})\) Please Register or Log in to view the hidden image!
Did you mean to say you think I am an internet hack? I am a little sensitive because of experience with people who have closed minds and the idea they are superior to those who have not been properly indoctrinated into their private club. However, Riemann could make the important break through he made, because was not properly indoctrinated and could think outside of the box. The show I watched connected the Riemann Zeta Function with Pythagoras' "music of the spheres" and I think the reason we have failed to prove the Riemann Function is the bias we are working with, that prevents us from thinking outside of the box. Pythagoreans connected math, music and astronomy (geometry) in a way that revealed cosmic harmony and this comes up in both the Riemann Zeta Function and Fibonacci sequence. However, if you do not think this worth our consideration, we can close our minds and this thread.
No, I wasn't implying that. Rather I was commenting that due to the basic nature of the Fibonacci series (add previous two numbers to get next) compared to the Zeta function (defined by analytic continuation in the complex plane using an integral representation) when a hack is looking for some mathematics to scatter through their nonsense they'll gravitate to the former rather than the latter. Pythagoras did a lot for his time but he didn't make the sorts of connections you're attributing to him in that manner. The principle of a right angled triangle is a pretty core one for geometry but it only gets you so far. As it happens Riemann developed an enormous generalisation of geometry which goes far beyond Pythagoras. From the time of people like Pythagoras and Euclid up until Gauss it was believed the only sort of valid geometry was that which satisfied the Euclidean geometry axioms, leading to the "Distance between a point (x,y) and (X,Y) is \(L = \sqrt{(x-X)^{2} + (y-Y)^{2}}\)". Gauss had developed a notion of hyperbolic geometry but was so appalled by its implications he delayed publishing it. Riemann developed what we now call Riemannian geometry by saying the combination of terms can vary as you move around a space. For simplicity let's say we want the length of the shortest path from the origin to some point (x,y), call it L. Pythagoras and Euclid say that \(L^{2} = x^{2} + y^{2}\) and the path is a straight line. But is this the only logically reasonable possibility? Nope. What if we didn't want to equally weight the two directions? What if that weighting varied? Riemann said that on a tiny scale any curved path looks straight (ie if you zoom in enough) so provided x,y are small, let's called these little line elements dx and dy instead, then we can consider a general weighted combination for the small piece of the path as \(dL^{2} = g_{xx}dx^{2} + g_{xy}dxdy + g_{yx}dydx + g_{yy}dy^{2} = g_{xx}dx^{2} + 2g_{xy}dxdy + g_{yy}dy^{2}\) (that simplification follows from the need for distance from A to B being the same as B to A). Now we work out the path length by adding up all the little bits, doing some calculus and you get an integral \(L = \int_{path}\sqrt{g_{xx}dx^{2} + 2g_{xy}dxdy + g_{yy}dy^{2}} = \int_{path}\sqrt{g_{ab}dx^{a}dx^{b}}\). For the Euclidean/Pythagorean case you have the simplest possible example of \(g_{xx}=g_{yy} = 1\) and \(g_{xy}=g_{yx}=0\). For hyperbolic geometry you have \(g_{xx}=g_{yy} = \frac{1}{y^{2}}\). This mathematics is the core of general relativity, pretty much all GR is is trying to work out what g is given a distribution of matter and energy via the Einstein field equations. So Riemann surpassed Pythagoras in geometry, never mind his Zeta function being more interesting than the Fibonacci sequence. You say you think the reason we've failed to solve the Riemann Hypothesis (RH) is because of closed minds and while I have more than enough grievances with the state of mathematical education and various academic practices I feel that you're not exactly basing that position on much information. Have you explored specifically what the RH is about, the specific formal definition of the problem? Given the nature and level of your questions I'd guess not. I am certain the eventual solution to the hypothesis will be incredibly imaginative, require a ton of entirely novel mathematics and a creative mind (or minds) but I don't believe it'll be overcome by people looking a bit more at the Fibonacci sequence in the manner often done by internet hacks. Various approaches to the RH have already drawn on many different domains, not just the obvious number theory (ie prime stuff) or function analysis (integral stuff) directions. No one in the research community is going to ignore other areas of mathematics which have applicability to a problem they work on just because of some tradition, a solution is a solution regardless of what label you want to put on the domain it comes from. As a general comment about internet mathematical hackery, all too often said hacks don't know what the mathematical community has done or is doing nor do they know much mathematics beyond high school so they are in no real position to be making the sorts of assertions about the community they often like to do. Yes, the Fibonacci sequence appears in a number of different areas and it links to all sorts of nice geometry results. I have yet to see a hack spouting "Look at all this amazing geometry involving circles and triangles and some primes, aren't I deep and insightful!" whose results don't fall into one of two categories: numerological BS or trivial implications of much more general results formally defined, explored, proven and used by the mathematics community already. The amount of number theory on right triangles, Pythagorean triplets and primes is enough to fill ten life times of work. Too bad internet hacks generally don't put in ten minutes of work.
