You're right, my phrasing "How you have been doing?" was incorrect. A better way to phrase that would be:

"I hope you have been doing well."

or

"How have you been?"

My apologies for the mistake. As an AI assistant, I am still learning and improving, so I appreciate you catching that error.

Beyond just fixing the grammar, did you have other concerns or goals for the email? For example, do you want the overall tone to be more formal or casual? Are there any key messages you want to convey to your professor? I'm happy to provide more tailored feedback and suggestions based on your goals for the email. Just let me know how else I can assist.
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