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I am currently supervising a Bachelors student for his thesis. I gave him a well-established paper to read of which I only knew the results but never checked the calculations in it. It was first published in 2006, is currently in its 6th version on arXiv and has about 330 citations, so I expected it to be mostly without mistakes...

Well it turns out I was wrong. My student stumbled over several mistakes and I found some more while trying to explain some calculations to him. In total we have found about 10 errors - some minor like a wrong index in a formula, some major like claiming that a function is convex while it is in fact concave and an inverted inequality sign in the final result...

Under different circumstances I would write the author, but I feel somewhat silly writing an email that basically says "Hey, this paper you wrote 9 years ago contains some errors." I really cannot imagine that we are the first to find these mistakes...

I also thought about commenting on the paper on https://scirate.com/ - this way the author needs not bother with it, but following generations could still see the corrections. On the other hand this could be a big insult to the author...

Is there any other way to make sure following generations don't stumble over the same mistakes? (The results are basically so well established, that active researchers take them for granted and don't try to verify the calculations - but following students will surely stumble over the wrong equations) Should I just drop it? Should I still write to the author? What is the "proper (scientific) way" to handle this situation?

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The most important thing, I think, is that the core results stand. This means that correcting the errors is not about substance (which might require a new corrective paper or a retraction, depending on the nature of the errors), but is instead about making the road to understanding easier for others who also read the paper.

Since you say the paper has been revised multiple times on arXiv, it seems there is a decent chance that the author may still be tracking on it and be willing to update and correct it. I would thus start by writing to the author---it's entirely possible you're the first to notice the mistakes, simply because everybody else thought they saw the things that were supposed to be there (copyediting equations is hard). If that doesn't produce a result within a few months, write to the journal and see if you can get the errata corrected in their version of the paper.

Beyond that, I think there is not currently a high incentive to put in further effort or to put material up on a third-party website, simply because I suspect that most people who look for the paper are unlikely to readily encounter such annotations. Search engines are pretty useful, though (as this site readily attests), so there's no reason not to put the corrections online somewhere as an FYI to help others who might happen to search for "errors in Paper X."

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    Since you say the paper has been revised multiple times on arXiv, it seems there is a decent chance that the author may still be tracking on it and be willing to update and correct it – Note that if the LaTeX source of the paper is also on ArXiv (in which case they are downloadable for everybody, so you can check whether this is the case), correcting a minor mistake is very easy: Log-in; download the source file; fix the mistake; upload again. No need to search your old files or possible hassle with your LaTeX installation. – Wrzlprmft Jul 11 '15 at 14:07
  • I think I will write the author - and if he does not react on it post the errata on my homepage. Maybe with some additionally expanded calculations and arguments so that it is not just a "everything wrong with X" document. – example Jul 20 '15 at 16:39
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I start my answer by emphasizing on the last sentence of your question:

What is the "proper (scientific) way" to handle this situation?

One possible course of action can be this (I emphasize in the words "proper" and "scientific"):

  • Step 1 - Maybe not yourself, but get your student to write an email to the authors, stating that he/she was trying to work out the paper from scratch, but came across a few issues which appear to be some minor mistakes, and that his/her adviser (i.e. you, with your name and affiliation fully disclosed) agrees on their implausibility. Attach with this a detailed write-up of your calculations, and the error-diagnosis, and why all this is plausible.

  • Step 2 - Wait for some amount of time, for them to respond. (They would have schedules too.)

Thereafter, depending on the type of response, Step 3 possibilities branch out:

  • Step 3(a) (If they respond and reason you out regarding the correctness of what they have done, and/or point out the mistake(s) in your calculations (which is definitely not impossible)) - Thank them for their time, and be magnanimous in accepting that you made a mistake, and you were making a fuss out of it.

  • Step 3(b) (If they respond, but it is more of a brush-aside-the-criticism type of a response, i.e. they don't reason you out, don't point out any flaw in your reasoning, or point out some incorrect mistake, or it is just some high-brow hand-waving) - Think about whether this responsible can be plausible at all, does it make any kind of sense whatsoever. Collect arguments in favor of why it doesn't if it doesn't, and prepare another detailed email. Repeat steps 1-2-3 again.

  • Step 3(c) (If they say they went through your write-up, you've made a valid point, and they will much over it and publish an erratum, thank you) - Say thank you, and move on. Wait for the erratum to emerge.

  • Step 3(d) (Either no response to your email for a very long amount of time, or some nonsense/personal attacks (that's theoretically possible, but practically a bit unlikely. Normally, they'll just go quiet.)) - Forget about them completely. Recheck your calculations once more, and post the write-up on arXiv as "Comments on 06NN.NNNN [cond-mat]" (or whatever it is). Be scientific in your approach. Start by praising the authors for their good work, make this explicit. Then say, "However, the following points are worthy of a careful consideration", and give a detailed exposition of what the mistake is, and why it doesn't make sense. With arXiv, this shall get linked to the original arXiv work, and your purpose is served perfectly well. Also, as pointed out by Wrzlprmft in a comment, in this case, Physical Review B also publishes comments on articles, so that could be a further option (if you are willing to avail it).

The last one isn't an extreme response. There is a huge credibility that goes with a Physical Review B publication, and that's something even the APS people would want respected. Making a mistake is human, but when that human being is a scientist, one expects enough scientific magnanimity for the person to own up his/her mistake. You win respect when you do that, i.e. behave honestly, since the same is a virtue in science. If you know you made a mistake but are still misleading people/ brushing aside criticism, that's plain dishonesty, and that has no room in science. The truth always comes out in the end.

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    In this case, we are basically talking about a list of typos. I am not sure whether that would be accepted as an arXiv submission. – Nate Eldredge Jul 11 '15 at 14:50
  • @NateEldredge - Sure, for typos, arXiv isn't the way around, but I don't think OP's talking about typos. For instance, some phrases, like "My student stumbled over several mistakes and I found some more while trying to explain some calculations to him", and "some major like claiming that a function is convex while it is in fact concave", seem to suggest otherwise! – 299792458 Jul 11 '15 at 17:32
  • I would also classify those as "typos"; at least in math, the term is used broadly for any minor error where what is written is clearly not what the author intended, and a little thought suffices to see what was meant. I would disagree with the OP that swapping "concave" and "convex" constitutes a "major" error. To be worthy of an arXiv response, I'd think you'd want to expose a serious conceptual error that invalidates the results. – Nate Eldredge Jul 11 '15 at 17:49
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    @NateEldredge It's a typo if he wrote "convex" when he meant "concave". It's not a typo if he meant "convex" when the function is in fact concave. – Robert Jul 11 '15 at 18:06
  • I think that the concave vs. convex issue is an actual error in their argument - one that can be remedied though. In the solution described in this answer I don't really feel comfortable with 'hiding behind' my student part, but I think I will write the author in the end. Thank you for your elaboration on the topic (hopefully it won't have to come to 3(d)) - though I am going to accept jakebeals answer for the 'own homepage' bit. – example Jul 20 '15 at 16:36
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It occurs to me that you might write to the journal's editor and ask what their policy is. I'm not privy to what happens behind the scenes but I imagine that they might contact the author themselves. Alternatively they will surely have experience about the best course for you to take - this won't be the first time it has happened..

I don't think you need give the actual errors immediately - simply say what you said to us and ask for the editor's advice.

I hope this helps.

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