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Recently I wrote a paper and submit it to a journal. Currently it's under review.

Later, after checking the preprint over and over, I noticed typos (for example instead of considering $\mathbb{N}^*$ I wrote $\mathbb{N}$) and I made modifications (not significant): grammar errors, improving the content.

During the review most of the above will be noticed.

If I was asked to make revision should I include the typos... ?

Also, is there any problem if an unnecessary remark - which isn't used and doesn't have an influence on the paper, it's remark of the form theorem x and theorem xx are true if we replace $P_1$ by $P_2$ a result which can be concluded trivially - is removed ?

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First, I would endorse the two answers already given. As far as the issue of removing superfluous or unnecessary material goes, I would be inclined to remove the material, but with provisos. First, before removing it, consider whether the material now seems superfluous to you because of your familiarity with the subject. It might not be considered superfluous by a reader who is less familiar with the material than you. Indeed, unless the material is simply annoying because it interrupts an otherwise clear chain of thought, I would be inclined to leave the material as it stands. If the reviewers found the material obtrusive, they would say so. The second proviso is that if you do delete the material, you should highlight the fact in the covering letter for the revision and make it clear that the material could be restored if necessary.

I've done this several times in resubmissions (both with mathematical papers and others), and have received both kinds of responses from editors along the lines of either "please restore the deleted lines", or "yes, I agree that the deletion makes the paper flow more smoothly". In one instance, I was asked to put my proposed deletion into an appendix, smoothing the flow of the main paper, but providing a supplementary support for some readers who wanted it.

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Typically, if you find any errors (even if the reviewers dont notice them), you should correct them during the review process. Regarding all the issues you noted, it should be possible to submit a revised version with your corrections and an answer to the reviewers in which you must include and comment on all changes to your manuscript.

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If I was asked to make revision should I include the typos... ?

Yes, you can and should make any appropriate edits when you submit a revision. Note what you changed, like any other. If you receive no requests for revisions, you can make these changes at the proof stage.

I'm not in math, so I won't address the second part.

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