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## Zeta and susy

Consider the separation of the Riemann Zeta function in two terms

\begin{flalign*} \zeta(s) &= 1^{-s} + 2^{-s} + 3^{-s} + 4^{-s} + 5^{-s} + 6^{-s} + ... = & \\ &=(1^{-s} + 3^{-s} + 5^{-s} + 7^{-s} + 9^{-s}+ ... ) + ( 2^{-s} + 4^{-s} + 6^{-s} + 8^{-s} + ...)&=& \\ &= (1 - 2^{-s}) \zeta(s) + 2^{-s} \zeta (s) &=& \zeta (s) & \end{flalign*}

which is pretty tautological, and now the same play with the Dirichlet Eta function,

\begin{flalign*} \eta(s) &= 1^{-s} - 2^{-s} + 3^{-s} - 4^{-s} + 5^{-s} - 6^{-s} + ... = \\ &=(1^{-s} + 3^{-s} + 5^{-s} + 7^{-s} + 9^{-s}+ ... ) - ( 2^{-s} + 4^{-s} + 6^{-s} + 8^{-s} + ...) &=& \\ &= (1 - 2^{-s}) \zeta(s) - 2^{-s} \zeta (s) &=& (1 - 2^{1-s}) \zeta (s) \end{flalign*}

The pair of functions $J_\mp \equiv \frac 12 (\zeta(s) \pm \eta(s))$ smells to susy quantum mechanics, doesn't it? Note how the pole (in s=1) of the Zeta function is cancelled by substracting both functions, and that the difference between $J_+$ and $J_-$ amounts to a zero in s=0.

Is this formalism used in number theory? Have the functions $J\pm$ some specific name?

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 Blog Entries: 6 Recognitions: Gold Member Now that th-phys.stackexchange questions are stable in phys.se, let me point to some extra context to this thread: http://physics.stackexchange.com/que...-number-theory also perhaps related, a curious twist with spin projections instead of susy: http://pseudomonad.blogspot.com.es/2...s-comment.html and rummiations on 24: http://blog.vixra.org/2011/02/28/the...-and-4-qubits/
 Recognitions: Gold Member We have for the Dirichlet Eta eta(s) = (1 - 1/(2**(s - 1))*zeta(s) (cf Derbyshire, Prime obsession, p 148)

Blog Entries: 6
Recognitions:
Gold Member

## Zeta and susy

 Quote by RamaWolf (cf Derbyshire, Prime obsession, p 148)
Also "Gamma", by Julian Havil. And I am a bit puzzled that the canonical text on the subject of Riemann Zeta Function, the one of H. M. Edwards, does not seem to find any use for this function.
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