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I am wondering if it is OK to send a finished manuscript to some mathematicians in your field for initial references or feedbacks in addition to just posting it in Arxiv, even if they do not know you completely, before submitting to a journal?

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    $\begingroup$ My advice: first have your adviser look it over; then post it to the arXiv (arxiv.org). $\endgroup$ Nov 25, 2021 at 4:07
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    $\begingroup$ It is OK to send the manuscript before posting it to the arXiv if (1) you have a strong reason to believe they would be interested in your work (you answered a question they asked, you improved on their result, you use crucially their work and their work has a small number of citations), and (2) you don't expect to get any reply or more than a few tiny comments. $\endgroup$
    – Will Sawin
    Nov 25, 2021 at 4:35
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    $\begingroup$ Yes. I can't see how this could not be OK. But of course you need to be polite and don't phrase it like you expect an answer. $\endgroup$ Nov 25, 2021 at 5:55
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    $\begingroup$ It's OK, but if you don't get a reply it is NOT OK to nag them. $\endgroup$ Nov 25, 2021 at 7:38
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    $\begingroup$ Many mathematicians don't regularly check arxiv (or other preprint servers). They either hear about new results from colleagues/students/conferences, or get preprints via mail/email. It is certainly okay to send preprints (or better yet, their arxiv links), and in many cases there is no other way to reach a person. $\endgroup$ Nov 25, 2021 at 14:52

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Informing specialists about a new manuscript posted on the arXiv is a common practice. Once it is posted there is no need to send a file.

But on my opinion, asking explicitly a person you do not know to read and comment is too intrusive and impolite. You can write something like this: "I would like to bring to your attention the new preprint... I would appreciate any comments". Or "I will be grateful for any comments".

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