"There are many bars downtown."
Translation:Det er mange barer i sentrum.
When referring to a city centre, can one say i sentrumet, or is it invariably i sentrum?
It depends. "I sentrum" refers to the fact that something is in a centre, "i sentrumet" that it is in a specific centre.
So could I say Det er mange barer i sentrumet to mean There are many bars in the centre? (If so, this answer should be accepted here.)
As a beginner: go for "i sentrum". I can imagine saying "Det er mange barer i bysentrumet", but it sounds kind of strange.
I would rather use "sentrumet" for talking about the center of a geometrical shape: "Linjene møtes i sentrumet (til sirkelen)" or "Linjene møtes i sirkelens sentrum".
downtown never takes the, but that doesn't mean it's not definite. There are many bars downtown means the same thing as There are many bars in the centre