You said your stairs had a carpet strip down the center, but how are the sides finished? Making some basic assumptions, I would do the following:
First, the stairs are not squeaking in the center, they are squeaking most probably on the sides, one or numerous of the four sides. Once you have isolated exactly where the squeak is originating, the most effective way is to countersink 4″ screws and screw the step down. The problem again, is how is the stair finished?
Should you find that the step is squeaking at the front or back edge (hopefully near the center), then you can remove the carpet, screw down the step and no finish is required as the carpet will then cover up the screw. Here you would make the screw hold flush with the floor.
Hope this helps,
Terry Alexander
