Fürstenberg’s proof on the infinity of primes revisited

August 6, 2010

Warning to teaching staff: summing infinite sequences of positive (>0) integers is difficult, but possible. For example \sum_{n=0}^\infty n \cdot n!=0 is not a mistake and students must get full marks. Let me show you why.

A couple of months ago I was hoping to ‘cheat’ a proof of the Erdös discrepancy conjecture by using a variant of an idea of Fürstenberg’s proof on the infinitude of primes. Remember, Fürstenberg considered the integers with a new topology. Its open sets  are \{a+n b: n\in\mathbb{Z}\} for a\in\mathbb{Z} and b>0.

This topology is metrizable. There are a couple of hand-waving arguments how this metric could look like. However, as far as I am aware of, there is so far no neat description in the literature. A couple of days ago, R. Lovas and I. Mezö have published a fairly straightforward proof that d(n,m)=\|n-m\| with \|n\|:=\frac{1}{\max\left\{k\in\mathbb{N}_{>0}:1|n,2|n,\ldots,k|n\right\}} induces the above topology.

Since \|n!\|\leq\frac{1}{n} the sequence (n!)_{n\in\mathbb{N}} converges to 0 in this topology. The partial sums of \sum_{n=0}^\infty n \cdot n! satisfy \sum_{k=0}^{n-1} k \cdot k!=n! and thus \sum_{n=0}^\infty n \cdot n!=0.

R. Lovas and I. Mezö have collected more such observations in their note. What they did not mention explicitly, but what I consider interesting is that with the above metric, the integers become an ultrametric space. Without loss of generality we assume \|m\|\leq\|n\|. Then 1,2,\ldots,\frac{1}{\|n\|} are all divisors of m and n and thus they are divisors of m+n. Therefore \|m+n\|\leq \|n\|=\max\{\|m\|,\|n\|\}. The strong triangle inequality now follows d(m,n)=\|m-l+l-n\|\leq \max\{d(m,l),d(l,n)\}.

Science and Responsibility – Love Parade 2010

August 2, 2010

Last week several hundred thousands of young people from all over the world gathered in Duisburg (Germany) to celebrate this years Love Parade.  21 of them are now dead. Their families and the public are mourning. What remains are questions. Questions on why this could happen and who is responsible.

Since this is a blog about science I am not so much interested in the  mayor, who refuses to step back since he wants to ‘clear things up’. I am also not interested in the organizer who claimed 1.400.000 participants and, maybe after he realized that this will get him into trouble with his insurance, counted again to just find 250.000. Police and fire services also played some role which is not of my concern.

I am interested in the security concept which was certified by a professor from a near university. No names here since this is an ongoing case and it is hard to get the facts. Sure is that the organizer planned to get more than 250.000 people through a 40 meter wide tunnel to the location. This tunnel was also planned to be the only exit from the location. At first glance that sounds crazy and … at a second glance that still sounds crazy. How could someone certify such a security concept? Our scientist in question has a Ph. D. in theoretical physics and his wiki entry contains some name dropping in form of nobel prize winning collaborators. If he signs such a concept then maybe because of some ‘deep’ insights stemming from his research on ‘transport and traffic’. However, this is buried in some proprietory journals to which I have no access (as an unaffiliated random guy).

That leaves me with the publicly available information. That is a TV interview right after the catastrophe with the event still going on. My rough translation (out of memory) of the decisive passage ‘… the behaviour of panicking people is hard to model …’ followed by some disgusting allocation of responsibility. In a further interview our scientist denies responsibility at all since he was not allowed to see the whole concept. This assertion is shockingly unmasking.

My conclusion: As long as results are not publicly available, as long as concepts  are signed without being fully seen and as long as no responsibility for mistakes is taken over it is very hard to differentiate between a charlatan and a scientist.

Because of the omnipresence of this case in all media science has lost a lot of reputation in germany.