Though it may not be a common problem, some Xperia Z1 users report problems of their touchscreen not responding. or freezing. You can test the responsiveness of the screen by doing this – Settings- About Phone –Diagnostics- Test Device-Touchscreen. Draw a line and see if it works.
It is seen that if there is a lot of activity done on the touchscreen for instance, playing a game or typing continually through the Swype keyboard, sometimes the touches do not materialize.
Usually, not touching the screen for sometime, gets the screen working fine. Else, restart the phone and you should be able to solve the issue.
Make sure that the phone is running the latest version of the firmware. This can be done by going to – Settings – About Phone-Software Updates – System Updates
If the touchscreen is not working is not working, after going to the diagnostics, checking the firmware and removing a few suspecting apps, you may have to get in touch with the Sony Service Center.