I have sent my manuscript to a professor of math about one month ago. This manuscript was written according to his encouragement for a joint research work. He told me that send him what I have done from my side, and then he will add additional materials from his own side. I want to send a tracking email about the status of that. Should I do that or I must wait for his reply?
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Do you really mean "tracking email"? Like these?: https://academia.stackexchange.com/questions/82869/is-it-rude-to-use-tracking-softwares-for-the-emails-that-you-send-to-potential-a Never, ever use these. – Bryan Krause Jul 15 '20 at 18:54
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1If by tracking email you mean an email sends you a notification when the recipient opens it, then it's probably a bad idea. – GoodDeeds Jul 15 '20 at 18:55
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Can you elaborate on the history? When did he tell you that, when was your last communication about it, how much does the joint work date back? – user151413 Jul 15 '20 at 19:17
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1The academic clock can run slow. Be aware. – Buffy Jul 15 '20 at 20:06
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In the non-Covid times, some researchers would take summers off, say June to August. In that case, you could not expect a reply until he returns to civilization in September. – GEdgar Jul 15 '20 at 21:47
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1@GEdgar True, but I have yet to find someone that goes into hibernation for three months like a bear and does not even answer e-mails. – Federico Poloni Jul 16 '20 at 13:03
2 Answers
"I want to send an email email asking him for an update about the status of that. Should I do that or I must wait for his reply?"
Absolutely. Why? Because it has been 1 month and you have not received any communication from him yet. Usually after waiting a full 7 days, it is not inappropriate to send a follow-up email.
"How much is an average time of waiting to recieve a reply from Professional co-author?"
It varies. Some people reply the same day, some the same hour, but it's also not uncommon for people to take a month or two to reply.
Usually it is not inappropriate to ask for an update 7 days after the first email. Another one after another 7 days can also often be considered okay. It's when you send more than 2 follow-up emails, or when you do not wait ~7 days between them, that it might start to be seen as "nagging", but even then that's not always the case. Please send him a follow-up email and ask us what to do if you still have problems after that. My guess is that he'll reply (since he asked you for the manuscript).

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When communication becomes difficult, when human sensitivity comes into play, don't write mail after mail: call your collaborator by phone. Maybe on phone you can get them to make a credible promise; maybe they will make you understand how overburdened they are right now; maybe they will enjoy a little chat with you; hopefully, your relation with them will grow through direct conversation.

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Calling collaborators on the phone without any prior arrangement or emergency seems horribly intrusive to me. – Arno Jul 16 '20 at 13:52