Paul Anka wrote the English lyrics, and the melodic structure we all know, yes. The original tune was from a French song called ‘Comme d’habitude’(As Usual) co-written by Claude Francois.
Paul heard the tune in 1966 while on holiday in South of France. He flew to Paris and acquired adaption, recording and publishing rights. Paul rewrote the lyrics with then familiar phrase ‘Me Generation’ in mind.
Yes it is not Sinatra’s song in many respects. He sang it but so have I, he just sang it a lot better.
The Sinatra Crime Family had nothing to do with producing this song.
Paul Anka also wrote the theme song for Johnny Carson’s Tonight Show.