Why not asking about updates of search process
I think there are only two cases where you should follow-up with the search committee:
- The time by which they said they would get back to you has passed.
- You have another opportunity that requires a time-sensitive response.
A while ago I read some good advice (somewhere on this site) about how to deal with the waiting period after a tenure-track interview:
As soon as you're done with the interview & visit, thank everyone who you've met, and then immediately move on with your life under the assumption that you did not get the job, i.e., search for the next opportunities, etc.
Two aspects:
It's unlikely that you will get any helpful answer. If they give you a precise timeline like "three weeks and a half", they certainly follow some clearly defined process, the outcome of which can only communicated after three weeks and a half.
By the mere act of asking, you deviate from the communication protocol that was agreed on, which could come off as slightly negative (annoying). It probably won't make or break your chances, but it's still a signal you might want to avoid sending.
Your best bet is to be patient. Focus on other activities and write more applications, which will help you put that particular position in perspective. It's not your only shot.