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)?

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.
