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Hi,
My Z3 went dead yesterday and I have not been able to get it back on with either soft resets or by using the button next to the SIM card slot. When I put it onto charge, the LED comes on until 'Sony' appears on the screen, at which point it restarts again. I've left it to charge for a few hours and it just keeps repeating this. Any help would be much appreciated.
Hi @MikeT1, I'm so sorry to hear about your phone. 
Could you please try and remove your SIM and/or SD cards from your phone and see if you are able to get the past this screen when powering on your phone.
If you still can't past the Sony logo without the phone restarting over and over I'd recommend performing a software repair using Xperia Companion. Note that this will delete all data stored on your internal storage and since you can't unlock your phone it won't be possible to create a backup.
Let me know how it goes!