I forgot recharging the HP Prime. The akku is now absolutely empty and the calculator don't start up any more. The display remains black. What can I do?

I don't know about the Prime specifically, but many digital tools with lithium batteries won't turn on until the battery has a certain minimum charge. For example my old phone, if I let it discharge completely, then I had to plug it to the charger for about 10 minutes before it would even turn on.
I'd just try leaving it plugged in for some time.

Simply plug in the charger and wait for half an hour until the voltage level is sufficient for powering up.

The Prime unes a rechargeable lithium-ion battery. The common standard for these batteries is a built-in circuitry that prevents discharge below a certain limit (and overcharge as well, for that matter) to protect the battery against damage. At that point the device is turned off and will not turn on again until a safe voltage level is reached again.

Dieter

Thanks for explaining that. Today in work I had a query about a device (not an HP calculator but using that type of battery) which we had supplied to a customer. Our test/development unit in the office refused to turn on until the charger had been on for about 5 minutes (their phone bill not ours!) Now we know why.

Li-Ion batteries do not appreciate being left below a certain threshold voltage , you can damage or at best wipe lifetime off the battery.
As mentioned there is circuitry built in to prevent use below that threshold, BUT there is also small self discharge and possiblly a small discharge from stuff they can't disconnect from the battery in this state.
So don't leave it in a discharged state.