Trim the thick layer of fat from the prime rib, leaving a ¼-inch to ½-inch layer.
Place the prime rib in a heavy roasting pan and let it sit out for 2 to 4 hours to come to room temperature.
Make compound butter by combining soft butter, black pepper, salt, cayenne pepper, and herbs de Provence.
Spread the butter mixture over the prime rib and season generously with more black pepper and salt.
Preheat the oven to 500 F. Calculate cooking time by multiplying the weight of the beef by 5, round up to next whole number.
Roast the prime rib for the calculated time, then turn off the oven and do not open the door for 2 hours.
Check the internal temperature; if undercooked, turn the oven to 325 F until it reaches 120 F.
Remove the prime rib, let it rest for 20 minutes, then slice and serve.
To make au jus, pour off excess fat from the pan, add water or beef broth, and bring to a boil while scraping drippings.