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I would like to know what is correct of the following:

He is famous in town.

He is famous in the town.

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New HopeHe is famous in town.
He is famous in the town.

These are both possible. "in town" is a set idiomatic expression; it typically refers to the town or city where the speaker lives, or the nearest town/city to where the speaker lives, that the speaker frequents or visits. In the south-east of England, "in town" traditionally means "in (central) London".

"in the town" can refer to any town that has been mentioned or is known from context.