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

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

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