How many Americans lost their lives in the D-Day invasion?

You would really have to re-define that question to get an answer you want.
On D-Day itself, 6th June 1944, 1,465 American troops lost their lives. D-Day was ONLY the 6th June. Subsequent days were known as D+1, D+2 and so on. The beaches were used for continued landings for almost a month before European ports came into use. Each day more American servicemen lost their lives, but this was not strictly D-Day if it happened after 6 June.