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I submitted my paper to an Elsevier math journal specialized in my research area on May 3rd. The paper is a bit long but on a fairly mainstream topic. The paper status is still "with editor" since then.
Because of my job situation and the job evaluation on the line, it is crucial for me that the paper is published promptly. I know that 17 days is normal for "with editor" status. But given the type of the journal and my situation, is it too early to ask for an update on the status or let the editor know that the time is very important for me? or it might hurt the process? If it is too early, how long more should I wait?

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Yes, it's probably too early. Time to review varies by discipline (I believe maths is particularly long, while medical science can be relatively quick). In any case, two/three weeks is too short for anything other than an editorial rejection. I wouldn't start chasing a paper until two or three months had passed with no notifications.

Some journals offer expedited review, but this is usually for papers with urgent findings (e.g. on Covid), not for career reasons. You could ask, but be prepared for the answer to be "no".

Reviewing is a volunteer activity (lots of problems with that, but anyway...) Finding reviewers takes time, and it's unlikely you would incentivise them to speed up by saying "it would be helpful for the author's career if you did this quickly".

Rdd
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  • Thank you for your response. I understand and agree with your points. My question was about the editorial part of the process; if explaining the situation could help expedite this part (Asking the reviewers, ...) or might sound pushy in the early stage, which it seems you think is the latter case. – Random visitor May 20 '22 at 17:02
  • It's unlikely that you can expedite this part of the process much. Waiting for reviewers to agree to review the paper will form most of the time at this point. Again, you can ask, and if you're polite it could help by a few days (e.g. move your paper up the pile for editorial review). – Rdd May 21 '22 at 11:27