iOS 11 have caused many inconviencee to the Apple iphone users. And Apple nees to seriously fix all the issues ASAP as this is losing trust of Apple customer in Apple iphone/iPad devices. One of such problem is that after updated iphone to iOS 11 (iOS 11.0.3) and tried to log in my Apple ID on my iphone 7 but it "could not Sign in " (there was a problem connecting to the server! but same Apple ID could log in my iPad 4 and iPad Pro and Mac . Even after multiple attempts the users are failed to login to Apple Id after iOS 11 upgrade.
Some users are unable to log into iCloud, iMusic or any any other app that is linked to my Apple ID from my iPhone. I don't know why just not working. Due to this login issue user can't buy in-app purchases because of it, or download new apps. It is just endless loop of logging into your Apple ID. Sometimes the error message would be, app store error message "could not sign in" "an unknown error has occurred".
To fix the issue of not able to login in Apple Id, iOS 11:
Check if your "emergency/backup email address" maybe the main email address of a different Apple ID? In my case it was, and I solved this issue (after hours trying) by logging in with the old/emergency mail address (which required a different password) and then logging out and logging in with the new one.
Please note in above steps you can also restore via iTunes.