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Hello,

a few years ago, I found on ArXiv an article in which the authors (I think they were at least two to write it) claimed to have proven that the non trivial zeros of the Riemann Zeta function were all simple using the concept of Riemann surfaces. But unfortunately, I just can't find it back. Does someone know if such a result has been published and widely accepted by the mathematical community? Thank you in advance.

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4 Answers

up vote 23 down vote accepted

This is widely open. Moreover, I think we will prove the Riemann Hypothesis much earlier than the simplicity of the zeros (if true). The latter is somehow much more accidental, the only reasonable argument I know in favor of it is "why would two zeros ever coincide"? Note, however, that some automorphic $L$-functions do have multiple zeros. If I recall correctly, even a Dedekind $L$-function can have a multiple zero at the center.

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though there are results like a certain percentage of the zeros are simple... –  shenghao Mar 28 '11 at 0:03
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I am no expert, but can a Dedekind zeta function vanish multiply if there is no Armitage/Serre phenomenon, with a root number of -1 for an Artin representation in the decomposition? The MAGMA L-functions handbook code has an example. magma.maths.usyd.edu.au/magma/handbook/text/1385#15208 –  Junkie Mar 28 '11 at 0:31
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@Junkie: "I am no expert, but can a Dedekind zeta function vanish multiply if there is no Armitage/Serre phenomenon"...That sounds suspiciously like a question an expert would ask! –  Pete L. Clark Mar 28 '11 at 2:50
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I let you guys sort this out. –  GH from MO Mar 28 '11 at 3:21
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For a Galois extension, the Dedekind zeta function factors as a product over all the Artin L-functions of the irreducible representations, with multiplicity equal to the dimension. Thus for dim>1 (ie non-Abelian Galois groups) the Dedekind zeta function is forced to have multiple zeros, not just at 1/2 but all the way up the critical line. All this goes back to Artin, much before Armitage/Serre. –  Stopple Mar 28 '11 at 15:26
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Please note in article - 0802.1764, Riemann Hypothesis may be proved by induction, by R. M. Abrarov and S. M. Abrarov - authors did not claim the proof of RH. They only suggested that induction procedure may be used for RH. This is a nice paper. It contains useful equations that were not known in number theory.

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Why is this downvoted? I do not know the details of the preprint in question, but I just had a brief look, and it appears to be correct that the authors of that paper do not claim (contrary to what one might assume from the title and the listing in the other answer) to prove RH (what they seem to do is to establish an equivalence of RH with another assertion, as do various papers). So, this comment/answer seems useful (Gregory cannot comment); if it is not for a more subtle reason it would be nice if this was pointed out along with the vote. –  quid Apr 2 '11 at 11:29
    
Here is the arxiv page: arxiv.org/abs/0802.1764 –  B R Apr 2 '11 at 17:28
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My apologies for including this article in my list. I was misled by the title, by the sentence "At least one of these identities may be applied to prove the Riemann Hypothesis by induction" in the abstract, and by the statement, "The Induction Procedure can be applied over and over again for further validation of (19). Hence the Riemann Hypothesis is justified" toward the end of the paper. But the last sentence of the article seems to say they have only found a condition which, if true, implies RH. So far as I can tell, the authors publish only in the arXiv. –  Gerry Myerson Apr 4 '11 at 0:04
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To the best of my knowledge, it is still an open question as to whether all the zeros are simple. If you could find that article....

For what it's worth, any number of "proofs" of the Riemann Hypothesis have appeared on the ArXiv. Here are a few (I've not included three more that were withdrawn by the authors).

1006.0381 The Riemann Hypothesis, Ilgar Sh. Jabbarov (Dzhabbarov)

0906.4604 A Proof for the Riemann Hypothesis, Ruiming Zhang

0903.3973 Concerning Riemann Hypothesis, Raghunath Acharya

0802.1764 Riemann Hypothesis may be proved by induction, R. M. Abrarov, S. M. Abrarov [EDIT: It appears that this paper does not actually claim a proof of RH - see Gregory's answer to the question (and my comment on Gregory's answer).]

0801.4072 The Riemann Hypothesis and the Nontrivial Zeros of the General L-Functions, Fayang Qiu

0801.0633 From Bombieri's Mean Value Theorem to the Riemann Hypothesis, Fu-Gao Song

0709.1389 One page proof of the Riemann hypothesis, Andrzej Madrecki

math/0308001 A Geometric Proof of Generalized Riemann Hypothesis, Kaida Shi

math/9909153 Riemann Hypothesis, Chengyan Liu

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Nice collection! I particularly liked "One page proof of the Riemann hypothesis" and "Riemann Hypothesis may be proved by induction". Here is a generalization: "One page induction proof of the Riemann Hypothesis AND the Twin Prime Conjecture". Can you beat that? Perhaps "Three-line proof that Peano Arithmetic is inconsistent"? –  GH from MO Mar 28 '11 at 1:13
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@GH Maybe it would be beaten by something like Zagier's title. Say something like "A One Sentence Proof Of The Riemann Hypothesis" =) –  Adrián Barquero Mar 28 '11 at 1:56
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@GH, did you notice that "One page proof of the Riemann hypothesis" is 17 pages long? –  Gerry Myerson Mar 28 '11 at 3:19
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Nope :-) I am sure 16 pages are for non-experts and then there is 1 page of beef. –  GH from MO Mar 28 '11 at 3:23
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I finally managed to find back the article I was talking about. Just click on the green link in the first message of the following link: link text

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For convenience, I'm adding a link to the original Arxiv page. The has been updated twice with respect to the version you link (no, apparently the main result wasn't withdrawn or amended): arxiv.org/abs/0911.5138 –  Federico Poloni Mar 28 '11 at 11:45
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It appears that this paper has been very recently accepted by the International Journal of Mathematics and Mathematical Sciences, as per hindawi.com/journals/ijmms/aip/985323 –  Gerry Myerson Mar 28 '11 at 23:27
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I don't claim to have read it carefully, but a quick glance through makes me very suspicious: the arguments seem to abstract somehow, and there are quite a lot of computer-generated pictures that look as though they may be intended as substitutes for rigorous proofs. Basically, I can't find the beef anywhere -- for instance, I don't see any sign of hard estimates. Coupling that with GH's comment, I am left thinking I can safely ignore this one unless the experts suddenly get excited about it. –  gowers Apr 2 '11 at 9:42
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Already the very beginning makes me very suspicious. They make a claim about Riemann surfaces and give two references. One is a page in Ahlfors book, where we find no theorem, but an informal discussion of Riemann surfaces. Ahlfors says on the previous page "This idea leads to the notion of a Riemann surface. It is not our intention to give, in this connection, a rigorous definition of this notion. For our purposes it is sufficient to introduce Riemann surfaces in a purely descriptive manner. (cont. in next comment) –  GH from MO Apr 2 '11 at 14:12
    
(cont. from previous comment) We are free to do so as long as we use them merely for purposes of illustration, and never in logical proofs." The second one is a conference proceeding publication by the authors which does not even show up in MathSciNet. Also, it does not look good on the second author that his 33 publications published over 40 years only have 4 citations in MathSciNet. The first author is 80 years old which raises further doubt about the credibility of the paper. In short: This paper is simply wrong by all likelihood. –  GH from MO Apr 2 '11 at 14:17
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