# Zeta and susy

#### arivero

Gold Member
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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#### RamaWolf

We have for the Dirichlet Eta

eta(s) = (1 - 1/(2**(s - 1))*zeta(s)

(cf Derbyshire, Prime obsession, p 148)

#### arivero

Gold Member
(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.

"Zeta and susy"

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