Jagex publishes no death count. A Reddit user who logged the in-game death broadcasts with RuneLite counted 3,114 deaths on the first day; OutputLag reported the tally with the deadliest monsters: giant spiders with 269, Moss Guardians with 146 and the jail guards with 144. It is an unofficial count of broadcasts, and the broadcasts were throttled the same afternoon, so the real number is higher.
Game Pulse reads the deaths it can verify itself: the official hiscores keep a death record for a dead Hardcore character, and the race page lists the recent ones with their date and world.