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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 :(

Jose Javier Garcia
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    I'd say that the correct prediction of experimental results accounts for the justification... – Danu Feb 01 '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 01 '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 01 '14 at 15:22
  • Related: http://physics.stackexchange.com/q/3096/2451 , http://physics.stackexchange.com/q/73066/2451 – Qmechanic Feb 01 '14 at 15:37
  • I think this should be of interest here. – Dilaton Feb 01 '14 at 16:45
  • http://physicsbuzz.physicscentral.com/2014/01/redux-does-1234-112-absolutely-not.html – Uncle Al Feb 03 '14 at 00:11
  • I really think that the fact that this works is a clue that we don't understand QFT all that well. I remember some book or another comparing QFT calculations to using algebra to calculate the volume of an irregular solid in the years before Newton -- possible, but frustrating, and not very elucidating.But all of the trick integrals and regularizations indicate to me, at least, that we are finding crutches that let us keep our formalism and get the right answer, but that we're not seeing the REAL formalism, which will make these mysteries clear. – Zo the Relativist Apr 28 '14 at 21:26

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