This physicist's treatment of the -1/12 issue from Zee's QFT book is the best treatment of the question I've seen.  
>Appallingly, in an apparent attempt to make the subject appear even more mysterious than it is, some treatments simply assert that the sum is by some mathematical sleight-of-hand equal to -1/12. Even though it would have allowed us to wormhole from (1) to (3) instantly, this assertion is manifestly absurd. What we did here, however, makes physical sense.