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 :(

  • 2
    $\begingroup$ I'd say that the correct prediction of experimental results accounts for the justification... $\endgroup$ – Danu Feb 1 '14 at 14:45
  • $\begingroup$ 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.] $\endgroup$ – Qmechanic Feb 1 '14 at 14:51
  • $\begingroup$ 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} $ $\endgroup$ – Jose Javier Garcia Feb 1 '14 at 15:22
  • $\begingroup$ Related: physics.stackexchange.com/q/3096/2451 , physics.stackexchange.com/q/73066/2451 $\endgroup$ – Qmechanic Feb 1 '14 at 15:37
  • $\begingroup$ I think this should be of interest here. $\endgroup$ – Dilaton Feb 1 '14 at 16:45

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