Riemann Rekindled
The demise of the arxiv continues into 2008 with yet another (cough) disproof of the Riemann Hypothesis (reported by Lubos). Elementary disproofs seem popular these days. Since Connes tells us the Riemann Hypothesis is closely related to Quantum Gravity, that means Quantum Gravity must be Elementary also. Elementary in the sense of axiomatically foundational, maybe?
Yesterday we came across categorified cardinalities once again. For example, to compute the cardinality of the groupoid of finite sets we just need to sum the cardinalities of the groupoid components,
|FinSet0| = $\frac{1}{1} + \frac{1}{2!} + \frac{1}{3!} + \frac{1}{4!} + \cdots = e$
The Riemann zeta function, for real arguments, looks a bit like such a sum, namely
$\zeta (s) = \frac{1}{1^{s}} + \frac{1}{2^{s}} + \frac{1}{3^{s}} + \cdots$
so when $s$ is a positive integer this might measure the cardinality of the sequence of products of cyclic groups $( \mathbb{Z}_{n} )^{s}$ for $n \in \mathbb{N}$. What sort of groupoid is this? It is very reminiscent of Rota's ideas on profinite combinatorics and the Riemann zeta function. Hmmm. We know that $\zeta (2) = \frac{\pi^{2}}{6}$ and so on, so the factors of $\pi$ must come from a cardinality for such a groupoid. The question is, what basic thing has (products of) cyclic automorphism groups? One possibility is oriented polygons and we already know that $n$-gons are associated with $n-3$ dimensional associahedra, and associahedra are related to the permutohedra, the vertices of which give the elements of the groups counted by $e$.
This seems like such a nice way to relate $e$ and $\pi$ and $-1 = e^{i \pi}$.
Yesterday we came across categorified cardinalities once again. For example, to compute the cardinality of the groupoid of finite sets we just need to sum the cardinalities of the groupoid components,
|FinSet0| = $\frac{1}{1} + \frac{1}{2!} + \frac{1}{3!} + \frac{1}{4!} + \cdots = e$
The Riemann zeta function, for real arguments, looks a bit like such a sum, namely
$\zeta (s) = \frac{1}{1^{s}} + \frac{1}{2^{s}} + \frac{1}{3^{s}} + \cdots$
so when $s$ is a positive integer this might measure the cardinality of the sequence of products of cyclic groups $( \mathbb{Z}_{n} )^{s}$ for $n \in \mathbb{N}$. What sort of groupoid is this? It is very reminiscent of Rota's ideas on profinite combinatorics and the Riemann zeta function. Hmmm. We know that $\zeta (2) = \frac{\pi^{2}}{6}$ and so on, so the factors of $\pi$ must come from a cardinality for such a groupoid. The question is, what basic thing has (products of) cyclic automorphism groups? One possibility is oriented polygons and we already know that $n$-gons are associated with $n-3$ dimensional associahedra, and associahedra are related to the permutohedra, the vertices of which give the elements of the groups counted by $e$.
This seems like such a nice way to relate $e$ and $\pi$ and $-1 = e^{i \pi}$.
posted by Kea | 11:40 AM
3 Comments:
The demise of the arxiv continues into 2008
Considering the way they have behaved, need we shed a tear?
"The demise of the arxiv continues into 2008 with yet another (cough) disproof of the Riemann Hypothesis (reported by Lubos). Elementary disproofs seem popular these days. Since Connes tells us the Riemann Hypothesis is closely related to Quantum Gravity, that means Quantum Gravity must be Elementary also. Elementary in the sense of axiomatically foundational, maybe?"
Connes paper on arxiv (an attempt to extend the standard model) a while back was filled with an enormous expanse of lagrangrian equations filling whole pages. He has got zero physical insight; he still couldn't make any falsifiable predictions or do anything with it that is really exciting. It's relatively easy to write down endless equations than to solve them and make connection to physical reality. That's of course regarded by some as just a slight difficulty in string theory.
When I tried submitting to arxiv in Demember 2002, I was hoping that people would read it and make constructive comments, but it was deleted in the few seconds between submitting it and reloading my web-browser in the library at Gloucester University.
Maybe the elite arxiv people are really brilliant geniuses who can spot errors without even checking papers. You have to respect their professionalism. It's really a pity that arxiv doesn't run the internet search engines like Google, and omit all crackpottery. Better still, there should be armed secret police paid to search out and destroy all non-mainstream idea papers, internet sites, and their creators. Then students wouldn't be confused about whether there are any alternatives to string theory. Kill off all alternatives first, then deny that they are possible. Good old totalitarian branewashing.
The Riemann hypothesis and rota's ideas are a nice way to relate e and pi, one that needs to be studied.
As we unfortunately saw last year, an arxiv controller isn't even a real scientist. From the one example, the main qualification is being a communist. Like Pinky and the Brain, they are still plotting to rule the world.
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