The digital display power is wired through the door switch, so it is intentional that it shuts off when the door is closed. If the display only intermittently comes on when the door is opened I would suspect a defective door light switch. Or if you are lightly touching the light switch and the display come on and goes off I would still suspect a defective light switch.
Prior to purchasing any parts, you should drop your control housing and inspect the wires that connect to the switch and control board. You can access the compartment with a thin flat blade screw driver. Insert the screw driver between the refrigerator liner and the control housing and very gently pry downwards on the housing. You will have to pry to the point that you think you are about to break the plastic and it will pop the locking tabs from the ceiling of the fridge. Start in the front right or left hand corners and work your way across the front first then when the gab is opened pull the side tabs free. There are two large tabs in the back that hold the back of the housing in place.
To make life easier you can un-plug the one large wire connector in the center of the control housing and remove the entire housing for examination.
The light switch part number is C3680310 (@ $10)
The control board part number is 12868513 (@ $120)
Hope this helps you out!
Let me know if you have any further questions!!