Zeta and susy

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arivero

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Consider the separation of the Riemann Zeta function in two terms

[tex]\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*}[/tex]

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

[tex]\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*}[/tex]

The pair of functions [itex]J_\mp \equiv \frac 12 (\zeta(s) \pm \eta(s))[/itex] 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 [itex]J_+[/itex] and [itex]J_-[/itex] amounts to a zero in s=0.

Is this formalism used in number theory? Have the functions [itex]J\pm[/itex] some specific name?
 
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arivero

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Last edited:
We have for the Dirichlet Eta

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

(cf Derbyshire, Prime obsession, p 148)
 

arivero

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(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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