Should I mention in my manuscript's cover letter that I am a PhD student? I am a PhD student in Mathematics. My adviser told me that I can submit our manuscript to some journal. So I am writing a cover letter and I am just wondering if I should mention in the cover letter that I am a PhD student.
If a cover letter is needed at all, its purpose nowadays is usually to include information for which there is no appropriate form in the journal’s submission system (which can inlude suggested reviewers, unwanted reviewers, informations about prior submissions, etc.) – see also these two questions.

I do not see how the information that you are a PhD student is relevant to the editor or anybody involved with the paper, so I wouldn’t mention it. Every spurious information in that letter just wastes the editor’s time. I submitted a few papers as a PhD student (with and without coauthors) without mentioning this fact and did not encounter any problems that I would even remotely relate to this.