How Do the Riemann Zeta and Dirichlet Eta Functions Interact?

[arivero]

Consider the separation of the Riemann Zeta function in two terms

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

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

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

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 canceled 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?

 

[arivero]

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/qu...g-theory-and-possible-link-with-number-theory

also perhaps related, a curious twist with spin projections instead of susy: http://pseudomonad.blogspot.com.es/2011/07/daniels-comment.html and rummiations on 24: http://blog.vixra.org/2011/02/28/the-24-cell-and-4-qubits/

 

[RamaWolf]

We have for the Dirichlet Eta

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

(cf Derbyshire, Prime obsession, p 148)

 

[arivero]

(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.