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The Riemann Hypothesis for High School Students: why proof so difficult
Dear All,
with reference to my previous posting:
The Riemann Hypothesis for High School Students,
some of you might have rightly wondered for what reasons its proof shall be so difficult.
Trying to approach such advanced topic in an elementary way, I attach a graph that may help you to get a mild intuitive hint.
Mind you, intuition in mathematics could yes be very informing about the various approaches that can be tried to solve a specific problem, but it often fights with absolute rigour ...
Anyway, if you look again at the Figure 1 of my previous posting, you will notice that the coordinates of the point of convergence are:
X = sum of all segments X components
Y = sum of all segments Y components
The graph herewith attached depicts some of the X components (cosine terms) for the case of said Figure 1. You can clearly recognise an alternating sign pattern. The Y components follow a similar pattern, but of a sine type.
Well, a non trivial zero corresponds to the sum of all such terms being simultaneously zero for both the X and Y components.
"Intuitively" that would already appear to be an amazing feat ... but it does happen, as billions of such zeros has so far been calculated (numerically).
The fact that the known non-trivial zeros have so far been calculated by numerical means only (there is no known elementary function for identifying said zeros) testifies to the inherent difficulty of the task.
We do not even know how to predict in terms of elementary functions the values of said zeros ... Try then to think for a moment for what strange reason said infinite sums of X and Y components have some faint chance to simultaneously be zero if, and only if, that operation on index n is the square root ... and you will have gained just a glimpse about the intricacies of such an endeavor.
Not pretending to be rigorous, of course.
Luca
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