Exploring the Separation of Riemann Zeta and Dirichlet Eta Functions

In summary, the Riemann Zeta function can be separated into two terms, with one term being a multiple of the original function and the other term being a multiple of the Dirichlet Eta function. The pair of functions J_\mp \equiv \frac 12 (\zeta(s) \pm \eta(s)) may have applications in susy quantum mechanics and number theory. The Dirichlet Eta function also has a connection to the Gamma function. However, the canonical text on the Riemann Zeta function does not seem to make use of this function.
  • #1
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 canceled 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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  • #3
We have for the Dirichlet Eta

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

(cf Derbyshire, Prime obsession, p 148)
 
  • #4
RamaWolf said:
(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.
 
  • #5


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.
 

Related to Exploring the Separation of Riemann Zeta and Dirichlet Eta Functions

1. What is the Riemann Zeta function?

The Riemann Zeta function is a mathematical function that was introduced by the German mathematician Bernhard Riemann. It is defined as ζ(s) = ∑n=1 1/ns, where s is a complex number. It is a very important function in number theory and has applications in physics, engineering, and other areas of mathematics.

2. What is the Dirichlet Eta function?

The Dirichlet Eta function is another mathematical function that was introduced by the German mathematician Peter Gustav Lejeune Dirichlet. It is defined as η(s) = ∑n=1 (-1)n-1/ns, where s is a complex number. It is closely related to the Riemann Zeta function and has similar applications.

3. What is the significance of exploring the separation of Riemann Zeta and Dirichlet Eta functions?

The separation of Riemann Zeta and Dirichlet Eta functions is a topic of interest in number theory and mathematical physics. It involves understanding the relationship between these two important functions and how they can be used to solve complex problems. It has implications in various fields, such as cryptography and signal processing.

4. How do the Riemann Zeta and Dirichlet Eta functions differ?

The main difference between the Riemann Zeta and Dirichlet Eta functions is in their definition. While the Riemann Zeta function sums the reciprocals of all positive integers, the Dirichlet Eta function sums the reciprocals of all odd positive integers. Additionally, the Riemann Zeta function is defined for all complex numbers except s = 1, while the Dirichlet Eta function is defined for all complex numbers with a positive real part.

5. What are some open questions in the exploration of the separation of Riemann Zeta and Dirichlet Eta functions?

There are many open questions in this area, including the relationship between the non-trivial zeros of the Riemann Zeta function and the poles of the Dirichlet Eta function, the existence of a functional equation relating the two functions, and the behavior of these functions on the critical line Re(s) = 1/2. Further research and exploration in this field may provide a deeper understanding of these functions and their significance in mathematics and other fields.

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