Mathematicians remain skeptical, and neither proof has been subjected to a serious analysis.Karl Sabbagh (2004). The Strange Case of Louis de Branges. London Review of Books, 22 July 2004. The main objection to his approach comes from a 1998 paper (published two years later)Conrey, J.B.; Li, Xian-Jin (2000) A note on some positivity conditions related to zeta and L-functions. International Mathematical Research Notices 2000(18):929–40 (subscription required; an abstract can be found here and a 1998 arXiv version here). by Brian Conrey and Xian-Jin Li, one of de Branges' former Ph.D. students and discoverer of Li's criterion, a notable equivalent statement of the Riemann hypothesis. Peter Sarnak also gave contributions to the central argument. The paper which, contrarily to de Branges' claimed proof, was peer-reviewed 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. 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 claimed 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 Karl Sabbagh, 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 said that he did not believe that the Riemann hypothesis was 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.
Mathematicians remain skeptical, and neither proof has been subjected to a serious analysis.Karl Sabbagh (2004). The Strange Case of Louis de Branges. London Review of Books, 22 July 2004. The main objection to his approach comes from a 1998 paper (published two years later)Conrey, J.B.; Li, Xian-Jin (2000) A note on some positivity conditions related to zeta and L-functions. International Mathematical Research Notices 2000(18):929–40 (subscription required; an abstract can be found here and a 1998 arXiv version here). by Brian Conrey and Xian-Jin Li, one of de Branges' former Ph.D students and discoverer of Li's criterion, a notable equivalent statement of the Riemann hypothesis. Peter Sarnak also gave contributions to the central argument. The paper which, contrarily to de Branges' claimed proof, was peer-reviewed 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. 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 claimed 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 Karl Sabbagh, 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 said that he did not believe that the Riemann hypothesis was 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.