You are correct, sorry for the confusion. The phrase “How you have been doing?” is not grammatically correct. The correct way to ask this question would be “How have you been doing?”

Here is the email with this mistake resolved:
```
Dear Professor,

Hopefully this email finds you well. I wish you a very merry Christmas and a happy holiday season.

I also wanted to let you know that I have successfully started my PhD program and everything has been going well so far.

How have you been doing? I hope that you have been enjoying the holiday season and that everything is going well for you.

Best wishes,
```
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