Monday, December 12, 2005

Riemann's hypothesis

I just received a lot of interesting snail mail. The first one is from Prof. Winterberg, one of the discoverers of cold fusion. He argues against the extra dimensions, using a picture of naked fat people (actually, some of them are M2-branes) and a German letter he received from his adviser, Werner Heisenberg. Very interesting but I apologize to Prof. Winterberg - too busy to do something with his nice mail and the attached paper.

A publisher wants to sell the 1912 manuscript of Einstein about special relativity. Another publisher offers books about the Manhattan project and Feynman's impressive thesis.

One of the reasons I am busy now is Riemann's hypothesis. Would you believe that a proof may possibly follow from string theory? I am afraid I can't tell you details right now. It's not the first time when I am excited about a possible proof like that. After some time, I always realize how stupid I am and how other people have tried very similar things. The first time I was attracted to Riemann's hypothesis, roughly 12 years ago, I re-discovered a relation between zeta(s) and zeta(1-s). That was too elementary an insight that was far from a proof but at least it started to be clear why the hypothesis "should be" true. The time I need to figure out that these ideas are either wrong or old and standard is increasing with every new attempt - and the attempts become increasingly similar to other attempts of mathematicians who try various methods. Will the time diverge this time? :-)

Arthur Jaffe indicates that he believes that he does not think that anyone can prove it, maybe except for Alain Connes, because it has not been proved for 150 years despite many attempts of ingenious people and it can't be proved now by mathematical induction either. ;-) I beg to differ. There are many new insights that were unknown to the great mathematicians in the past, and the problem simply can be cracked at some moment. The zeta function and primes have appeared in p-adic string theory but this research direction was always something like a collection of identities than an actual theory whose physics could imply important insights. The new approach may be more promising.


  1. Riemann Hypothesis can be proved since the Riemann Xi function is the determinant of a Hamiltonian operator with a certain potential

    see for example :

    the Riemann-Weil trace formula (distributional) is just the trace of Tr(d(E-H) here d(x) is a delta function

  2. Riemann xi function is the determiannt of a certain Hamiltonian operator

  3. Sorry, if I understand well, your Hamiltonian is the usual QM kinetic energy p^2/2 plus potential energy and the potential energy is related to the zeta function itself again, right? So it's circular reasoning because something will fail to work if RH is false, won't it?

  4. it depends on the argument of the Riemann xi function which can be calculated by Riemann Siegel without evaluating any primes or Riemann zeros

  5. Dear Jose, please don't be silly. Riemann xi function is, up to trivial rational factors, powers of pi, and a gamma function, the very same thing as the Riemann zeta function, so it has the same nontrivial zeros.

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