my phone 5s turns on then it goes to a blue screen then it shuts off and does it all over again

same for me

This almost always means there is hardware damage or the phone was jailbroken. You can try restoring in iTunes after placing in recovery mode, but there's a good chance it won't work and you'll have to replace the phone.

it is a professional iphone. I never jailbroke it. I tried several things: reset, restore and full reset. After a full reset, it gets better for a few days and then again the same "blue screen of death" and very frequent reboots. The fact it gets better for some time is what makes me think it could be software and not hardware. Apparently, I'm not the only one to get that issue.

Define "full reset." If you restored to factory settings and this recurs, the problem is hardware related unless illegal software was installed.

I'm also unclear why you think that other's having this issue has any bearing on the cause or solution.

full reset = restore to origin with recovery mode.

OK, then I guess this is hardware. No illegal software has ever been installed.

I don't know whether the fact that others have exactly the same issue means that it is the same root cause. I just observe the issue which is very painful. If you can help me finding the cause and solution, I would greatly appreciate.

The only way to determine the cause and solution is for the phone to be evaluated by Apple.