Suppose that after submitting a paper for publication, but before hearing anything back from the referee, I discover an error in the paper that needs to be corrected, or an omission that needs to be rectified. What is the best course of action? Should I send a revised version to the editor to forward to the referee, explaining exactly what I have changed? Or wait to hear back from the referee and then include the fixes in my next version along with whatever other changes the referee suggests (again explaining them to the editor)?
[Math] How to correct an error in a submitted paper
journals
Related Solutions
Some advice explicitly directed at less senior people. I would very much advise some who does not yet have tenure to NOT take the nuclear option (e.g. posting a paper on the arXiv accusing someone of being wrong, or writing irate letters to the editors of a journal). In the extremely rare cases in which this has to be done, it is best done by someone who is both pretty senior and very politically skilled. This leads me to my other piece of advice. Namely, talk to other, more senior people in your research area. First, they might be able to convince you that it isn't really as serious an error as you think. Second, they will probably know the personalities involved better, and be more effective at convincing an author to do the right thing if something has to be done.
The two times something like has happened to me, I had ended up proving stronger results than the erroneous papers by pretty different techniques. I buried remarks at the ends of the introductions of my papers mentioning the wrong papers and explaining where they went wrong. On one of those occasions the author had left math and I didn't know how to contact him, so I didn't correspond with him first (after I posted the paper the arXiv, one of his friends contacted him and we exchanged some friendly emails). The other time, I explicitly cleared the language I used with the original author.
1) Has this happened to anyone else? Is this a relatively common occurrence, or am I just sloppy?
It is not very common (the usual preventive techniques include showing the draft to a few experts/friends, putting it on ArXiV, letting it lie for a month or two and then rereading it, etc. before sending it to a "top" journal) but it happens now and then. What is common is severe difficulty with finding an error in one's own work. I would say that affects more than a half of mathematicians I know. The reason is that you read not what is written but rather what you believe should be there when you just finish typing the manuscript and start proofreading. The main trick of good proofreading is to turn yourself into a complete idiot, who doesn't see a single step ahead, has no idea of the overall structure of the argument, takes everything literally, and is not convinced of anything that is not clearly put in a modus ponens form. Needless to say, it is about as hard for a shrewd person to read like that as for a genuinely stupid one to read between (or over) the lines. And even if you know all that, you are still destined to submit or even publish papers with errors. Just a few months ago, I was informed about an error in one of mine published papers. It was just a remark and the statement was actually correct, but the proof wasn't. So, to have this kind of public shame now and then is almost inevitable whether you are an unknown postdoc, or Andrew Wiles, or something in between. I'm not sure if Poincare published a single formally correct proof in his lifetime and people still are completely puzzled by some passages in Linnik's works, so you are in a good company.
2) The anonymous referee is probably someone distinguished in my field. Do they now have a bad impression of me? (This probably is not a question that can easily be answered . . . .)
It is actually easy to answer: most likely, for him you are nobody, your name is just a random combination of letters, and your "initial value" is zero. An erratic paper leaves it this way, so nothing is lost. We are all getting worthless papers to referee every month and I challenge everyone to recall the name of the author of some bad paper he rejected 6 months ago. The only scenario in which "someone distinguished" would bother to take a mental record of your name after looking at a single opus of yours is when he finds something interesting and unusual in your work. Then your value for him is currently positive, though, of course, not as high as it would be if you solved the problem. So, again, there is absolutely nothing to worry about.
3) If I manage to patch up this paper, is it reasonable to resubmit it to this journal, or have I burned my bridges there?
Of course, it is. What matters is not how many mistakes you made on the road and who saw them but whether you finally reached your destination and whether other people consider that destination worth reaching. The theorem and its correct proof lose nothing in value if somebody published or tried to publish 20 false proofs before that. That some of those false proofs might be proposed by a person with the same name and biometrical characteristics as those of the one who finally found a correct proof changes nothing in the grand scheme of events. So, if you manage to fix the error and make sure that the argument is, indeed, correct, I see absolutely nothing wrong with submitting it again because from a purely logical standpoint, it is a different paper. If you get it returned solely on the grounds that the previous version was incorrect and not based on the merit considerations (even correct and good papers get rejected sometimes for various reasons), it'll merely tell you that the jornal is not really "top" but just "snobbish", in which case I would avoid it altogether in the future (at least, until they change the editorial board).
Best Answer
I think Angelo is right in broad strokes. For minor errors it's annoying to bother the editor and the referee. But if it's something that the referee might end up wasting hours on then correcting it pre-emptively makes sense. Here are some examples of things that I'd wait or not on for concreteness. I'd love to hear from people with more knowledge/experience about whether this is vaguely the right place to draw the line.
Wait when:
Contact immediately when: