Learn English – How you can say “Happy New Year” formally

There are many ways to express this idea. The one I use most often at work when I would like someone to feel free to ask me questions is:

Please don't hesitate to contact me if you have any questions or concerns.

Another way you might express it is:

Please let me know if you have any questions.

To make it more polite you would request that the reader contact you by using "please"; Omitting the "please" sometimes makes it seem like a demand instead of a request. In the right context something like "Let me know if you have any questions." is fine. The "please" just makes sure the tone comes across as polite.

Also, if you want to be very deferential to someone like a professor, you could imply that the reason they might have a question is due to some fault of yours, for example you didn't explain something clearly or you may have misunderstood their message. For example,

Please ask if something I've written isn't clear.

I think the weirdness comes from the fact that this is not really a conversation, in the sense that it is not an exchange of information between people who intend to be talking to each other.

In some cultures (including the US, where I am), it is considered friendly to acknowledge someone entering (and leaving) your store with some kind of greeting. Asking after their health or well-being is a formula; practically no one expects you to answer the question in its actual meaning.

This is part of a set of formulas that I think practically every culture has for conversation in one way or another. There are papers written on the subject. In this case, perhaps you could think of it as the speaker's way of letting you know that they have noticed you and are free to talk with you; you can reply with any formula of your own ("fine", "ok", "better than a sharp stick in the eye") to acknowledge that they have spoken; if what you say sounds enough like a formula itself (the first two, not the last), then you will not be expected to continue a conversation.

But if you came into the shop looking for something you wanted help with, the fact that they said something might help you know they were free to listen to what you wanted at that time, in case you would otherwise have been shy about asking.