The official Main Theme is "Welcome to Los Santos" by Oh No.
Though, quite honestly, I wish it was this;
In case you didn't notice, it's a modernised remix of the GTA: San Andreas theme song. It uses a more modern west coast hip hop style than the classic 90's gangsta rap era which evolves a lot more than the simple 32 bar sections used for the SA theme. I think it is a nice and clever evolution from GTA: San Andreas and cheered me up when I first started the game.
I think it's a mix of III and SA's themes. listen to both and then listen to Welcome to Los Santos.