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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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    $\begingroup$ 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. $\endgroup$ Commented Sep 8, 2010 at 12:51
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    $\begingroup$ 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!! $\endgroup$
    – Zen Harper
    Commented Sep 9, 2010 at 1:22
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    $\begingroup$ 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 ... . $\endgroup$
    – Emerton
    Commented Sep 10, 2010 at 18:42
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    $\begingroup$ 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. $\endgroup$ Commented Sep 12, 2010 at 16:34
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    $\begingroup$ Given the comments in this thread, I see this as a quintessential subjective and argumentative question. I don't think that either the math community or MO participants on its behalf owe Zen Harper an explanation on why have de Branges' papers on RH not been published or his ideas not been reviewed in print. In spite of modifying the formulation to satisfy formal criteria, OP's true purpose seems to be advancing an agenda based on persecution complex or finding a justification for it. See in particular comments from Sep 9 at 1:22, Sep 12 at 13:45 and Sep 12 at 13:55. $\endgroup$ Commented Sep 12, 2010 at 18:18

2 Answers 2

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The paper by Conrey and Li "A note on some positivity conditions related to zeta and L-functions" https://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. Lagarias, Jeffrey C., Hilbert spaces of entire functions and Dirichlet $L$-functions, Cartier, Pierre (ed.) et al., Frontiers in number theory, physics, and geometry I. On random matrices, zeta functions, and dynamical systems. Papers from the meeting, Les Houches, France, March 9–21, 2003. Berlin: Springer (ISBN 978-3-540-23189-9/hbk). 365-377 (2006). ZBL1121.11057, MR2261101. Author's website and Wayback Machine.

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    $\begingroup$ Note that Li was a student of de Branges (graduated in '93). $\endgroup$ Commented Sep 8, 2010 at 14:13
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    $\begingroup$ 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. $\endgroup$
    – Zen Harper
    Commented Sep 9, 2010 at 0:15
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    $\begingroup$ The link to the paper by Lagarias has expired. What was the paper? (In general, I feel that links to articles in MathOverflow posts should for preference be accompanied by written journal references for precisely this reason.) $\endgroup$
    – Ian Morris
    Commented Mar 9, 2016 at 18:09
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    $\begingroup$ @IanMorris I suspect that the paper by Lagarias is "Hilbert spaces of entire functions and Dirichlet $L$-functions," in Frontiers in Number Theory, Physics, and Geometry I: On Random Matrices, Zeta Functions, and Dynamical Systems, pp. 365-378. As of this writing, there is a copy of the paper on Lagarias's website. $\endgroup$ Commented Nov 11, 2021 at 17:50
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    $\begingroup$ I have included the full citation - as mentioned above, the link to Springer website doesn't work at the moment. (Although the same broken link is on MathSciNet website.) $\endgroup$ Commented Jan 14, 2023 at 13:51
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In the years since this question was first asked and answered, there have been some new developments (namely, 80+ pages of commentary by Eric Kvaalen, and a large number of new versions of de Branges' paper).

Starting with the Wiki (which provides a decent account of events), I followed a link to Eric Kvaalen's "Commentary on work of Louis de Branges" from 2016, a 70-page paper, available only in the very unorthodox format of PNG files. The relevant versions of de Branges' paper can be found here: https://web.archive.org/web/20160701000000*/http://www.math.purdue.edu/~branges/proof-riemann.pdf, from 2013-2016 (past 2016 the links are dead). I will type out Kvaalen's conclusion (from page 70), regarding I think this version of de Branges' paper (archived on Nov. 17 2015):

In this wide-ranging paper, de Branges has enunciated many true statements but also quite a few false statements or statements with no proof offered. Having analyzed the first 59 of his 90 pages, I will stop at this point because the later portions of what I have analyzed do not provide a good foundation for the rest. Besides which, de Branges has now replaced this version with a new version (dated November 23, 2015).

Linked in the aforementioned Kvaalen webpage is a slightly later paper (also 2016), "The application of the de Branges method to prove the Riemann Hypothesis" regarding a much earlier version of de Branges' paper from 2006. The versions from 2004-2008 can be found here http://web.archive.org/web/20080901000000*/http://www.math.purdue.edu/~branges/riemannzeta.pdf (past 2009 the links are dead).

(Looking only at the first page, there seems to be at least 5 eras of de Branges' paper: 2004 where he began with a section on "Locally compact skew-fields"; May 2005 where he began with a section on "Hypercommplex analysis"; Sep. 2005-2008 where he began with a definition of a de Branges space; 2013-2014 where he began with a section titled "The Inverse Problem for the Vibrating String"; and 2015 onward where he begins with a section titled "Generalization of the Gamma Function". And that's only looking at obvious differences on the first page…)

Anyways, I retype the introduction and some conclusions that Kvaalen had about the 2006 version:

… in 2005 [de Branges] wrote a 41-page paper called "A Proof of the Riemann Hypothesis", the last version of which is dated July 2006. Around 2007 he started on a somewhat different line of attack, and he removed the 2006 paper from his website. His own opinion on his older argument has wavered. As recently as June 2014 he told me that the older argument was correct, but in conversations in May 2017 he was unable to recall exactly what he had meant in a key sentence, and thought that there was not a proof in the paper.

I have analyzed a recent version of his current work [above mentioned "Commentary …"] and found it wanting. But in this paper I would like to go back to his earlier work. The 2006 paper is hard to follow and does not explain all his reasoning. It presents three theorems, but does not even show how they can be applied to the Riemann Hypothesis. I will show how to prove the Riemann Hypothesis using Theorem 3 in it, if that theorem is true.

Unfortunately, the last sentence of his attempted proof of Theorem 3 is the key step which remains unexplained. There are also errors in the proof of Theorem 1 which I do not go into here. Theorem 1 may not be true, but here may be a form which is true that covers the specific case needed.

And from Section 5 "Conclusion regarding the Riemann zeta function":

I have demonstrated how it would be possible to apply the 2006 draft by de Branges to prove the Riemann Hypothesis, assuming Theorem 3 to be true. But the last sentence in de Branges's attempt to prove Theorem 3, “The computation of adjoints denies $w-ih$ as a zero of $E'(z)$ distinct from $\overline w$ or as a double zero equal to $\overline w$”, is problematic. It is not clear what exactly he meant, and whether the meaning is true. There is also doubt about the truth of Theorem 1. My analysis of the proof (which I am not yet ready to put on line) finds errors. However, de Branges has suggested to me that a weaker form would be true and sufficient. Theorem 1 is not actually mentioned in the rest of the paper, but there is one place where it seems to be used.

The version found on his webpage, which I presume is the most current one, seems to date from 2017: https://www.math.purdue.edu/~branges/proof-riemann-2017-04.pdf. His webpage also has his most recent attempts on the invariant subspace problem (105 pages), and a handwritten 13-page scan on the "Banach measure problem". I haven't seen any discussion on these papers, which although sad, seems justifiable based on the above complaints.

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