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Hi,

I know this is a dangerous topic which could attract many cranks and nutters, but:

According to Wikipedia [and probably his own website, but I have a hard time seeing exactly what he's claiming] Louis de Branges has claimed, numerous times, to have proved the Riemann Hypothesis; but clearly few people believe him. His website is:

http://www.math.purdue.edu/~branges/site/Papers

but I find his papers difficult to follow. However, whether or not you believe him, his arguments presumably should prove something, even if not the full RH.

So, my question is:

Are there any theorems related to the Riemann Hypothesis and similar problems, arising from his work, which have been fully accepted by the mathematical community and published (or at least submitted)?

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11  
First, I vote for this to stay open, though it might benefit from a bit more editing (but not by me). Second, I've heard that Lagarias looked at the approach some years back. Presumably it was found wanting then. –  Steve Huntsman Sep 8 '10 at 12:51
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I took the liberty of editing out the second question, which asks about what people believe. What remains seems to me an interesting and appropriate question, and I hope it can remain open. –  Pete L. Clark Sep 8 '10 at 17:38
14  
Hmmph!! Closed!! I don't see why it's "subjective" or "argumentative" - if de Branges is really the only person who believes it and everyone else disbelieves, that's not really "subjective" I think. If others believe, I want to know who! One mathematician claims to have proved RH, and almost all other mathematicians remain silent. What is going on here?! When I did a Google search, I found almost all stuff was written by journalists or other people with little mathematical understanding. How can Mathematics progress in this fashion? Have the courage to express your opinions!! –  Zen Harper Sep 9 '10 at 1:22
16  
Dear Zen, It is not a question of having courage. Rather, mathematics is a profession in which the practitioners are sticking their necks out time and time again: claiming the proof of a previously unsolved problem is always a gutsy thing to do, there is always the possibility of coming a cropper, and it is always painful if one's claim does in fact collapse. For this reason, people in the profession are always reluctant to be publicly critical of other's work, even if they are unsure about it. There but for the grace of God ... . –  Emerton Sep 10 '10 at 18:42
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Zen, let me preface my comment by pointing out that (a) I am far from an expert and (b) I am not a neutral observer (we are in the same dept.). You are of course free to give the silence any interpretation you wish. My own is perhaps a bit less dark: that it has more to do with apathy rather than conspiracy. So if you are strongly interested in this area, perhaps you can start reading through this stuff and ask technical questions here as and when you get stuck. –  Donu Arapura Sep 12 '10 at 16:34
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1 Answer 1

up vote 38 down vote accepted

The paper by Conrey and Li "A note on some positivity conditions related to zeta and L-functions" http://arxiv.org/abs/math/9812166 discusses some of the problems with de Branges's argument. They describe a (correct) theorem about entire functions due to de Branges, which has a corollary that certain positivity conditions would imply the Riemann hypothesis. However Conrey and Li show that these positivity conditions are not satisfied in the case of the Riemann hypothesis.

So the answer is that de Branges has proved theorems in this area that are accepted, and his work on the Riemann hypothesis has been checked and found to contain a serious gap. (At least the version of several years ago has a gap; I think he may have produced updated versions, but at some point people lose interest in checking every new version.)

Update: there is a more recent paper by Lagarias discussing de Branges's work.

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12  
Note that Li was a student of de Branges (graduated in '93). –  Thierry Zell Sep 8 '10 at 14:13
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Thanks very much. But that paper is from 1998; is there anything more up-to-date? I think he's made more recent updated claims and "proofs" around 2004 and 2009. Although, as you say, it's easy to see why people are reluctant to spend too much time checking his stuff in detail; especially since his writing style is not the clearest to follow. –  Zen Harper Sep 9 '10 at 0:15
    
I've accepted this answer since it's reasonable enough and probably the best we can hope for. –  Zen Harper Sep 19 '10 at 10:02
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