# can we PHYSCALLY (not by mathematics) justify that $\zeta (-s)= 1+2^{s}+3^{s}+4^{s}+…$

the values $\zeta (-1)= -1/12$ and $\zeta (-3)= 1/120$ give accurate results for casimir and to evaluate the dimension in bosonic string theory

so is there a physcial JUSTIFICATION to justify that in phsyics (not mahthematics) every time we see a divergent series like $1+2^{s}+3^{s}+.........$

otherwise how it would be possible that zeta regularization gave only correct results for the series $1+2+3+4+5+.....$ and $1+8+27+64+..$ but not for example for $1+4+9+16+25+..=0$

why a matheamtical fucntion would be useful only for certain values but not for others :(

• I'd say that the correct prediction of experimental results accounts for the justification... – Danu Feb 1 '14 at 14:45
• Comment to the question (v1): Presumably OP means that the parameter $s$ should be real (and $s>-1$ for it to be non-trivial). [Also $s=-1$ is excluded since the zeta function has a pole there.] – Qmechanic Feb 1 '14 at 14:51
• but could we rely only in experimental prediction to conclude that every divergent zeta regularizable series in physics satisfies $\zeta ((-m)= \sum_{n=1}^{\infty}n^{m}$ – Jose Javier Garcia Feb 1 '14 at 15:22
• – Qmechanic Feb 1 '14 at 15:37
• I think this should be of interest here. – Dilaton Feb 1 '14 at 16:45