My Game Center has been loggedout automatically and I cannot log in how to solve this

Gosh, just had the same problem, fought with it for some days, and finally solved it thanks to the internet. So I'll share the solution with you, hoping it'll fix the bug as it did by me.

First, go to the airplane mode in the preferences and activate it. Then, quit every active app by double clicking the home button and ejecting every running application. Then, do an hard restart pushing simultaneously the home and lock button and letting them click for some seconds: your iPhone should restart.

Very important: keep the airplane mode ON, and then open the GameCenter preferences; first sucess, it should open! Disconnect your account, turn the airplane mode OFF and be sure being able to access the internet again. Go to the GameCenter preferences and log in (inform your needed password). Now, GameCenter should be fixed!!!