most recent 30 from http://mathoverflow.net 2013-05-23T01:44:29Z http://mathoverflow.net/feeds/question/38049 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/38049/what-exactly-has-louis-de-branges-proved-about-the-riemann-hypothesis Zen Harper 2010-09-08T12:06:42Z 2010-09-12T13:01:50Z

<p>Hi,</p> <p>I know this is a dangerous topic which could attract many cranks and nutters, but:</p> <p>According to Wikipedia [<em>and probably his own website, but I have a hard time seeing exactly what he's claiming</em>] Louis de Branges has claimed, numerous times, to have proved the Riemann Hypothesis; but clearly few people believe him. His website is:</p> <p><a href="http://www.math.purdue.edu/~branges/site/Papers" rel="nofollow">http://www.math.purdue.edu/~branges/site/Papers</a></p> <p>but I find his papers difficult to follow. However, whether or not you believe him, his arguments presumably should prove <em>something</em>, even if not the full RH.</p> <p>So, my question is:</p> <p><em><strong>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)?</em></strong></p>

http://mathoverflow.net/questions/38049/what-exactly-has-louis-de-branges-proved-about-the-riemann-hypothesis/38057#38057 Richard Borcherds 2010-09-08T13:50:14Z 2010-09-09T00:41:32Z

<p>The paper by Conrey and Li "A note on some positivity conditions related to zeta and L-functions" <a href="http://arxiv.org/abs/math/9812166" rel="nofollow">http://arxiv.org/abs/math/9812166</a> 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. </p> <p>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.)</p> <p>Update: there is a more recent <a href="http://www.springerlink.com/content/l7n6670q03k54150/" rel="nofollow"> paper by Lagarias</a> discussing de Branges's work. </p>