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?
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".