I am refereeing my first paper and I'm quite excited! But inexperienced and I would like to ask an advice to the Maths Community of MO. Let me tell you that I have already read http://mathoverflow.net/questions/36596/refereeing-a-paper, but it seems that my question is quite different. Roughly:
What is the point after which you get nervous while refereeing a paper?
I've found many English mistakes. (Well, I'm not a native English and so I can understand. So I am not nervous yet)
I've found some maths inaccuracies like "let A be any set"... and then I have discovered that the proof of the first result works only for finite sets. (OK, those are only inaccuracies - I am not nervous yet)
There are many references like "we use the notation of [X]", "this result is proved in [X]", where [X] is a BOOK, without specifying a precise section, or the number of the result... should I get this book and read all to find out the correct references? - just thinking of it, makes me a bit nervous..
(most importantly). There is a mathematical more serious mistake. Something that might be fixed, but not obviously (my definition of obvious is three evenings, in this case). I am not saying that the paper is completely wrong but that now... now I'm getting nervous!
Now, taking into account that the person who asked me to referee this paper told me: be selective, we accept only 20% of submitted papers,
what should you do in these cases? Reject? Ask for a revision? Not getting nervous and try to see if the rest of the piece is good (I'm quite a good guy and I'm doing that at the moment)?
Of course I will talk with the editor, but I also would like to know more opinions that might be helpful in future.
Update: Thank you very much to everybody for the numerous and helpful comments.