Are there any good websites for hosting discussions of mathematical papers? I was wondering if there are any websites out there which 


*

*systematically provide space for the discussion of mathematics articles (particularly those on the arXiv, though not necessarily just those), and 

*have a large enough user base to have some hope of having a good discussion (or at least have hope of attracting one in the near future).


Of course, MO is not so far from such a site, but is organized around questions, not papers, which leads to rather different discussion in practice.  Similarly, some blog posts and nLab pages serve this purpose, but are not created systematically.  
 A: Slightly facetiously, I'm mildly tempted to vote to close as a duplicate.  However, I shan't and shall merely point out that most of the answers to the question: Is a free alternative to MathSciNet possible? are relevant to this question, particularly mine.  The question Errata database? which covers similar ground.
(The rForum mentioned in the latter has stalled a little.  It could be revived if sufficient interest is shown.)
A: There's a useful list of such sites at http://www.quora.com/What-are-some-websites-where-one-can-post-commentary-and-reviews-of-academic-papers; the best of the lot, in my opinion, is:
Annotatr
http://annotatr.appspot.com/
Try it out!
A: I am developing arXiv Analytics which offers more features and a better user interface for reading arXiv eprints. For each arXiv paper there is an accompanying review page which can be used for posting reviews or discussions. The website is still under development. Any feedback is greatly appreciated.
A: (This should be put in the thread right below the original question as it does not provide any answer, but I can't leave comments). Yan LeCun wrote down some ideas for a new publishing model/system for Computer Science, also talking about discussing/commenting papers. Reading the comments to this question, his suggestions seem quite relevant.
A: In Peer review 2.0 thread, from 2021, Martin Sleziak suggested to me this ten years old question. So, there is Papers$^\gamma$ with source code under CC0 Public Domain Dedication. It gains the popularity slowly, in an answer to another MO question I explain possible reasons behind this and give more examples of online platforms devoted to paper discussions.
Disclaimer: I'm the founder of http://papers-gamma.link.
A: It seems appropriate to mention Gowers's "Modest Proposal" from fall of 2011. Here is his original post, and then his "more modest proposal." This came about after the OP asked this question and seems very related, since a big benefit of his recommended mathoverflow-like system of submitting papers was the ability to have discussions of those papers. As others have pointed out, this also allows the community to quickly correct errors in the papers and also would reduce the lag-time between when new results are finished and when others can start to build on those results. I'm not sure what ever happened to Gowers's idea, except that there was a lot of discussion at the time and problems were pointed out. If someone knows, I hope he/she will edit this and fill in the details. This also seems related to Alex Chernov's recent answer; maybe Furmaniak's site was based on Gowers's ideas. I'm not really familiar with Reddit, so I don't know how similar it is to what Gowers envisioned. 
A: PaperHive seems to fulfill both of your requirements, as far as I have seen. It provides a discussion channel directly next to the document (or in a seperate tab) to ask questions and comment on marked test passages, or to ask for clarifications. 
Since arXiv is open access, one can look at most of the arXiv articles on PaperHive, as they are synced with PaperHives frontend. Just search for something, you can even use arXiv ID or DOI for some journals that are available. Most other things look self-explanatory.
A big counter-argument against PaperHive as a discussion board for arXiv articles is that there seems to be a lack of availability of arXiv articles newer than 2016. Maybe that is because of the lack of interest, but it is hopefully easy to fix. I just discovered this problem while writing this.
I'm surprised no one here has heard of it before, it has been around for four years now. It just has no big user base of mathematicians. I still see potential in the concept, and a lot of code for PaperHive has been made open source.
A: Time to mention the Selected Papers Network:  https://selectedpapers.net
EDIT:
I hope that Tim does not mind me editing this to add extra links:
Introductory discussion by John Baez: http://johncarlosbaez.wordpress.com/2013/06/14/the-selected-papers-network-part-2/
Introductory discussion by Tim Gowers: http://gowers.wordpress.com/2013/06/16/the-selected-papers-network/#more-4954
A: For papers having enough overlap with physics, one possibility is to use the Reviews section of PhysicsOverflow (note that it is possible for the registered PO users to submit there papers for review).
A: I'm pretty sure the answer to the question as asked is "No".  At present there does not seem to exist a unique web location dedicated to discussing each individual mathematical article.
It would be technically feasible, and to my mind advantageous, to set up a "talk" page attached to each arXiv article.  I have seen this suggestion on various mathematical blogs (unfortunately, I can't remember which ones, and googling doesn't turn them up; this answer, like the question, is CW, so if you know of any, please just include a link).  I certainly do not know enough about web design to draft such a system, and the folks who run the arXiv do, but are horribly overworked as it is, so I doubt that they'll be writing anything anytime soon.  I encourage others with massive web-fu to create some code and send it towards the arXiv administrators, but they might not even have the time to look at it.  (When you are thinking about how best to set up such a site, please do keep in mind that well-working online forums tend to have fairly heavy moderation.  Moderator attention is definitely not sufficient to have a well-run site, and not strictly necessary, but it helps.  Since there aren't nearly enough people out there eager to moderate random columns on the arXiv, one proposal I've seen is for arXiv authors to have the option to open a talk page, with the commitment that they themselves participate and moderate.)  (Another thing to think about is how to convince NSF to fully underwrite the costs at arXiv.)
Note that although most mathematics papers are available on arXiv, it is not the only database of papers.  I could imagine adding a talk page to each bibliographic entry at Numdam, for example, or to each article at Web of Science.
Something like a "talk" page does exist for (almost) every (published) mathematical article, but I don't think it's what the question is after.  MathSciNet and Zentralblatt include reviews of every (almost) article in their databases, written by experts.  This does not accomplish the goal of "systematically provid[ing] space for the discussion of mathematics articles", but it is something.
A: Although no such system exists formally (as far as I know) there is an informal way to do this. Namely make some comments on a blog or web page and then add a trackback to the arXiv page. If this happened on a regular basis then it could be developed into something more systematic along the lines you are suggesting. My feeling is that only a small percentage of papers would attract any discussion.
A: http://arxaliv.org/comments 
http://arxaliv.org
Ralph Furmaniak created a site based or reditt where people can leave comments and up-vote the arXiv papers. See discussions at 
http://publishing.mathforge.org/discussion/83/frontend-to-the-arxiv-httparxalivorg-based-on-reddit/
http://arxaliv.org/r/help/comments/12hd/welcome_to_arxaliv/is
I think this is great, unfortunately not so many contributors, let us contribute ? 
A: I think this is a very reasonable question.  It comes up often enough and deserves an answer.  In fact, I see not one but two potential forums.  Here is my take.  
1) Rumors, followup ideas, small errors, minor comments and remarks on the paper.  This is all worth doing on your own blog space.  I see no need to make this kind of discussion "official".  In fact, the diversity of blogs and opinions is a plus here: less "intimidation by experts", fewer worries about "degeneration" of the discussion, etc.  Finally, typos and mistakes in the paper are responsibilities of the author - when you find them, email them to her/him and stop worrying about other readers.  In short, I see no need for a single forum for a discussion of this kind. 
2) Serious comments, substantial remarks, delicate technical problems with the paper, etc.  I think a blog or wiki type discussion forum is a too informal/unserious for contributions of this kind.  On a positive side, there is a perfect forum for this kind of discussions: it's called the arXiv!  Remember, the arXiv was never meant to be a only a "free storage" of published papers.  It has moderators, calls uploaded papers "submissions", and once approved, calls them "publications".  A great feature of arXiv is that you can store and advertise there the kind of work that you don't intend to publish in traditional journals.  So if you have something important to say to everyone, don't be shy, follow Mnёv's example and post it on the arXiv (here are some other examples). Make sure to add hyperlinks to older arXiv papers if you want it to look more webby.  Finally, there is already a great tool to explore forward arXiv citations.  So why bother inventing a new forum when the one we have works fine, if only people used it more often for this purpose.  
A: You can try SciRate (https://scirate.com/), a site that allows you to rate and comment papers from Arxiv (it updates the list of papers automatically). It doesn't seem to be very popular, though (lack of critical mass?).
A: Wikiversity has useful technical features (the same as Wikipedia): support for mathematical formulas (MathJax), referencing tools, collaborative editing, discussion pages, version control, email alerts, etc. It can very well be used for discussing mathematical papers. 
