D=10?

I can find a lot of material telling that the regularized sum

1 + 2 + 3 + 4 + 5 + 6 + 7 + ... = - 1/12

Implies that D=26 in bosonic strings, via (D-2)/12 = 1

I would expect then that the equally well known sum

1 - 2 + 3 - 4 + 5 - 6 + 7 - ... = 1/4

shall imply, via (D-2)/4 = 1, that D=10

But a random exploration of the internet bibliography fails here... I
can not find any post using this argument. Is it too naive? Or too
well known to even mention it?

Alejandro
.