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Alexandre Eremenko
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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".

Alexandre Eremenko
  • 91.8k
  • 9
  • 260
  • 431