359:, who in 2003 had written a book on the Riemann Hypothesis centered on de Branges, quoted Conrey as saying in 2005 that he still believed de Branges' approach was inadequate to tackling the conjecture, even though he acknowledged that it is a beautiful theory in many other ways. He gave no indication he had actually read the then current version of the purported proof (see reference 1). In a 2003 technical comment, Conrey states he does not believe the Riemann hypothesis is going to yield to functional analysis tools. De Branges, incidentally, also claims that his new proof represents a simplification of the arguments present in the removed paper on the classical Riemann hypothesis, and insists that number theorists will have no trouble checking it. Li and Conrey do not assert that de Branges' mathematics are wrong, only that the conclusions he drew from them in his original papers are, and that his tools are therefore inadequate to address the problems in question.
355:) which de Branges would use to construct his proof would also force it to assume certain inequalities that, according to them, the functions actually relevant to a proof do not satisfy. As their paper predates the current purported proof by five years, and refers to work published in peer-reviewed journals by de Branges between 1986 and 1994, it remains to be seen whether de Branges has managed to circumvent their objections. He does not cite their paper in his preprints, but both of them cite a 1986 paper of his that was attacked by Li and Conrey. Journalist
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L-functions, a group even more general than
Dirichlet L-functions (which would imply an even more powerful result if his claim was shown to be correct). As of January 2016, his paper entitled "A proof of the Riemann Hypothesis" is 74 pages long, but does not conclude with a proof. A commentary on his
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That original preprint suffered a number of revisions until it was replaced in
December 2007 by a much more ambitious claim, which he had been developing for one year in the form of a parallel manuscript. Since that time, he has released evolving versions of two purported generalizations, following
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The particular analysis tools he has developed, although largely successful in tackling the
Bieberbach conjecture, have been mastered by only a handful of other mathematicians (many of whom have studied under de Branges). This poses another difficulty to verification of his current work, which is
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Meanwhile, the apology has become a diary of sorts, in which he also discusses the historical context of the
Riemann hypothesis, and how his personal story is intertwined with the proofs. He signs his papers and preprints as "Louis de Branges", and is always cited this way. However, he does seem
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yields several claims of proofs, some of them by mathematicians working at academic institutions, that remain unverified and are usually dismissed by mainstream scholars. A few of those have even cited de
Branges' preprints in their references, which means that his work has not gone completely
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and published in a scientific journal – gives numerical counterexamples and non-numerical counterclaims to some positivity conditions concerning
Hilbert spaces which would, according to previous demonstrations by de Branges, imply the correctness of the Riemann hypothesis.
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of the
Bieberbach conjecture was not initially accepted by the mathematical community. Rumors of his proof began to circulate in March 1984, but many mathematicians were skeptical because de Branges had earlier announced some false (or inaccurate) results, including a claimed proof of the
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largely self-contained: most research papers de
Branges chose to cite in his purported proof of the Riemann hypothesis were written by himself over a period of forty years. During most of his working life, he published articles as the sole author.
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independent but complementary approaches, of his original argument. In the shortest of them (43 pages as of 2009), which he titles "Apology for the Proof of the
Riemann Hypothesis" (using the word "apology" in the rarely used sense of
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in July 2008. It was retracted a few days later, after several mainstream mathematicians exposed a crucial flaw, in a display of interest that his former advisor's claimed proofs have apparently not enjoyed so far.
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Mathematicians remain skeptical, and neither proof has been subjected to a serious analysis. The main objection to his approach comes from a 1998 paper (published two years later) authored by
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Born to
American parents who lived in Paris, de Branges moved to the US in 1941 with his mother and sisters. His native language is French. He did his undergraduate studies at the
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unnoticed. This shows that de Branges' apparent estrangement is not an isolated case, but he is probably the most renowned professional to have a current unverified claim.
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in 1964 (incidentally, in December 2008 he published a new claimed proof for this conjecture on his website). It took verification by a team of mathematicians at
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Actually, the correctness of the Bieberbach conjecture was not the only important consequence of de Branges' proof, which covers a more general problem, the
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to validate de Branges' proof, a process that took several months and led later to significant simplification of the main argument. The original proof uses
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also gave contributions to the central argument. The paper – which, contrarily to de Branges' claimed proof, was
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in 1984, now called de Branges's theorem. He claims to have proved several important conjectures in mathematics, including the
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The Riemann hypothesis is one of the deepest problems in all of mathematics. It is one of the six unsolved
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interested in his de Bourcia ancestors, and discusses the origins of both families in the Apology.
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A note on some positivity conditions related to zeta and L-functions.
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He has released another preprint on his site that claims to solve a
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2000(18):929–40 (subscription required; an abstract can be found
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Li released a purported proof of the Riemann hypothesis in the
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Leroy P. Steele Prize for Seminal Contribution to Research
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In June 2004, de Branges announced he had a proof of the
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Controversial Claims of Solutions to Unsolved Problems
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543:[0807.0090] A proof of the Riemann hypothesis
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519:International Mathematical Research Notices
417:In 1989, he was the first recipient of the
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486:"Commentary on work of Louis de Branges"
272:and innovative tools from the theory of
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502:. London Review of Books, 22 July 2004.
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500:The Strange Case of Louis de Branges
428:In 2012, he became a fellow of the
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484:Kvaalen, Eric (January 14, 2016).
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466:A proof of the Riemann Hypothesis
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595:Mathematics Genealogy Project
575:"Louis de Branges de Bourcia"
430:American Mathematical Society
258:invariant subspace conjecture
188:(1953–57). His advisors were
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146:(born August 21, 1932) is a
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270:hypergeometric functions
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167:West Lafayette, Indiana
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651:Mathematical analysts
498:Karl Sabbagh (2004).
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600:Papers by de Branges
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569:O'Connor, John J.;
392:de Branges function
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525:and a 1998
332:Xian-Jin Li
312:L-functions
233:Diophantine
159:Mathematics
106:Mathematics
71:Nationality
610:Categories
454:References
243:theories.
217:functional
55:1932-08-21
448:Peter Lax
309:Dirichlet
266:Leningrad
529:version
469:Archived
436:See also
305:apologia
241:operator
237:spectral
225:harmonic
593:at the
403:measure
229:Fourier
221:complex
209:analyst
231:) and
102:Fields
527:arXiv
384:arXiv
364:arXiv
320:Hecke
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247:Works
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