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could I theoretically erase all the information on the internet?
No, because the majority of it is on non-volatile storage (magnetic discs and EEPROM aka Flash)
Some things might temporarily look slightly different when you reboot because parameters for dynamically generated content which were stored in RAM would be lost, and they would only regenerate the same way if the user interactions which set the parameters were replicated.
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One possible issue is if servers depend on checking in with other servers to start up correctly. There's going to be some interesting circular dependencies that no one's thought about or hopes will never happen.
Plus if you kept the whole internet off for a year, every server would have expired SSL certificates and might not be able to download new ones without manual intervention.
(there's an analogous problem with the power grid called a black start )
If a) every server in world was connected to a single power source; b) there were no back ups or off grid servers (or any other loop holes); and c) I turned the power off and on again, could I theoretically erase all the information on the internet?
As I suspect the answer is no, if I left it powered off for a year then turned it back on, would there be any issues (purely related to the information stored on the internet, not societal collapse)?