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.
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.
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 Here are a few (I've not included three more that were withdrawn by the authors).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
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