Exploring the Separation of Riemann Zeta and Dirichlet Eta Functions

[arivero]

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 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.

 

[blue_raver22]

I find this exploration of the separation of the Riemann Zeta and Dirichlet Eta functions to be intriguing. The tautological nature of the first equation is interesting, and it is fascinating to see how the same play can be applied to the Dirichlet Eta function.

The introduction of the functions J_\mp, which are essentially a combination of the two original functions, does indeed have a similarity to supersymmetry in quantum mechanics. It is also interesting to note the cancellation of the pole in s=1 when subtracting the two functions and the resulting zero at s=0 in the difference between J_+ and J_-. This suggests a deeper connection between these functions and the properties of the Riemann Zeta and Dirichlet Eta functions.

In terms of the use of this formalism in number theory, I am not familiar with any specific applications. However, it is possible that this approach could provide new insights and connections in this field. As for the specific name of the functions J_\pm, I am not aware of any specific terminology, but they could potentially be referred to as "supersymmetric combinations" of the Riemann Zeta and Dirichlet Eta functions.

Overall, this exploration raises interesting questions and connections between these functions and supersymmetry in quantum mechanics. It would be worth further investigation and analysis to fully understand the implications and potential applications of this formalism in number theory and other fields.