The title of the song is "Theme From New York, New York" which was a Martin Scorcese film starring Robert DeNiro and Liza Minelli. Yes, Liza's version is the original, but Sinatra did what Hendrix did to Dylan with All Along The Watchtower and (to a lesser extant) Joe Crocker did to the Beatles with With A Little Help From My Friends and blew the original away.