A hypothesis enunciated in by L. Bieberbach [1] : For all functions of class which are regular and univalent in the disc and which have the expansion in this disc, one has the estimate , only for the Koebe functions where is a real number. Bieberbach proved his conjecture for. The problem of finding an accurate estimate of the coefficients for the class is a special case of the coefficient problem. At the time of writing the validity of the Bieberbach conjecture had been established for. It was first proved for in by K.

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Rumors of his proof began to circulate in March , but many mathematicians were skeptical because de Branges had earlier announced some false results, including a claimed proof of the invariant subspace conjecture in incidentally, in December he published a new claimed proof for this conjecture on his website. In June de Branges announced he had a proof of the Riemann hypothesis , often called the greatest unsolved problem in mathematics, and published the page proof on his website.

That original preprint suffered a number of revisions until it was replaced in December 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 independent but complementary approaches, of his original argument.

In the shortest of them 43 pages as of , which he titles "Apology for the Proof of the Riemann Hypothesis" using the word "apology" in the rarely used sense of apologia , he claims to use his tools on the theory of Hilbert spaces of entire functions to prove the Riemann hypothesis for Dirichlet L-functions thus proving the generalized Riemann hypothesis and a similar statement for the Euler zeta function , and even to be able to assert that zeros are simple.

In the other one 57 pages , he claims to modify his earlier approach on the subject by means of spectral theory and harmonic analysis to obtain a proof of the Riemann hypothesis for Hecke 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 , his paper entitled "A proof of the Riemann Hypothesis" is 74 pages long, but does not conclude with a proof.

Peter Sarnak also gave contributions to the central argument. Specifically, the authors proved that the positivity required of an analytic function F z 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 and , 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 paper of his that was attacked by Li and Conrey. He gave no indication he had actually read the then current version of the purported proof see reference 1.

In a 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 released a purported proof of the Riemann hypothesis in the arXiv in July He signs his papers and preprints as "Louis de Branges", and is always cited this way.

However, he does seem interested in his de Bourcia ancestors, and discusses the origins of both families in the Apology. 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 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. The Riemann hypothesis is one of the deepest problems in all of mathematics. It ranks among one of the six unsolved Millennium Prize Problems. A simple search in the arXiv will yield several claims of proofs, some of them by mathematicians working at academic institutions, that remain unverified and are usually dismissed by mainstream scholars. An entire function satisfying a particular inequality is called a de Branges function.

Given a de Branges function, the set of all entire functions satisfying a particular relationship to that function, is called a de Branges space.

He has released another preprint on his site that claims to solve a measure problem due to Stefan Banach. Awards and honors[ edit ].

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## A simple proof of the Bieberbach conjecture

Rumors of his proof began to circulate in March , but many mathematicians were skeptical because de Branges had earlier announced some false results, including a claimed proof of the invariant subspace conjecture in incidentally, in December he published a new claimed proof for this conjecture on his website. In June de Branges announced he had a proof of the Riemann hypothesis , often called the greatest unsolved problem in mathematics, and published the page proof on his website. That original preprint suffered a number of revisions until it was replaced in December 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 independent but complementary approaches, of his original argument. In the shortest of them 43 pages as of , which he titles "Apology for the Proof of the Riemann Hypothesis" using the word "apology" in the rarely used sense of apologia , he claims to use his tools on the theory of Hilbert spaces of entire functions to prove the Riemann hypothesis for Dirichlet L-functions thus proving the generalized Riemann hypothesis and a similar statement for the Euler zeta function , and even to be able to assert that zeros are simple. In the other one 57 pages , he claims to modify his earlier approach on the subject by means of spectral theory and harmonic analysis to obtain a proof of the Riemann hypothesis for Hecke 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.

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## Louis de Branges de Bourcia

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## ド・ブランジュの定理

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## Bieberbach Conjecture

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