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.

3$\begingroup$ Not that this answers your question, but a multiple zero is in some sense even less likely than one would expect if the zeros were "randomly distributed" (given their average density): they seem to repel each other, so if you plot the zero spacings scaled to average 1 then their density approaches $0$ as the distance approaches $0$. [I hedge with "in some sense" because even random spacing would make the probability of a coincidence $0$. NB I'm trying to minimize confusion by using "zero" for a root of $\zeta(s)$ and "$0$" for the smallest nonnegative real number.] $\endgroup$– Noam D. ElkiesApr 29, 2016 at 3:03
3 Answers
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.

5$\begingroup$ though there are results like a certain percentage of the zeros are simple... $\endgroup$– shenghaoMar 28, 2011 at 0:03

4$\begingroup$ 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 Lfunctions handbook code has an example. magma.maths.usyd.edu.au/magma/handbook/text/1385#15208 $\endgroup$– JunkieMar 28, 2011 at 0:31

35$\begingroup$ @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! $\endgroup$ Mar 28, 2011 at 2:50

4

10$\begingroup$ For a Galois extension, the Dedekind zeta function factors as a product over all the Artin Lfunctions of the irreducible representations, with multiplicity equal to the dimension. Thus for dim>1 (ie nonAbelian 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. $\endgroup$– StoppleMar 28, 2011 at 15:26
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 LFunctions, Fayang Qiu
0801.0633 From Bombieri's Mean Value Theorem to the Riemann Hypothesis, FuGao 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

7$\begingroup$ 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 "Threeline proof that Peano Arithmetic is inconsistent"? $\endgroup$ Mar 28, 2011 at 1:13

4$\begingroup$ @GH, did you notice that "One page proof of the Riemann hypothesis" is 17 pages long? $\endgroup$ Mar 28, 2011 at 3:19

5$\begingroup$ Nope :) I am sure 16 pages are for nonexperts and then there is 1 page of beef. $\endgroup$ Mar 28, 2011 at 3:23

3$\begingroup$ 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. $\endgroup$– GregoryApr 2, 2011 at 5:06

2$\begingroup$ Here is the arxiv page: arxiv.org/abs/0802.1764 $\endgroup$– B RApr 2, 2011 at 17:28
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

1$\begingroup$ 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 $\endgroup$ Mar 28, 2011 at 11:45

1$\begingroup$ 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 $\endgroup$ Mar 28, 2011 at 23:27

3$\begingroup$ 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 computergenerated 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. $\endgroup$– gowersApr 2, 2011 at 9:42

1$\begingroup$ 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) $\endgroup$ Apr 2, 2011 at 14:12

1$\begingroup$ @gowers, @GH, I wonder whether it would be appropriate for you to convey your misgivings to the editors of the journal in question. $\endgroup$ Apr 4, 2011 at 0:11