If Z(-1) = 1+2+3+4+5+....= -1/12. & Z(0) = 1/1^0 +1/2^0 +1/3^0 +1/4^0 +...
Z(0) = 1+1+1+1+1+...
[Z(-1) +Z(0)] = 2+3+4+5+... = 1+2+3+4+5+... -1= -1/12 -1 = -13/12
This implies Z(0) = -1
[Z(-1) - Z(0)] = 0 + (1+2+3+4+5...) = -1/12
This implies Z(0) = 0
Both these values for Z(0) of -1 & 0 are contradictory.
Adding [Z(-1) + Z(0)] + [Z(-1) - Z(0)] = -13/12 -1/2 = -14/12 = -7/6
This implies 2Z(-1) = -7/6 => Z(-1) = -7/12This contradicts Z(-1) = -1/12 which
was the original assumption.
Putting infinite divergent series equal to a finite number appears to lead to
results that are not sensible.