Dirichlet Eta Function (Alternating Zeta Function)

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The Dirichlet Eta Function (sometimes called the zeta alternating series or the Euler zeta function) is defined by the following equation:

Connections to Other Functions

The Dirichlet Eta function is an alternating sign version of the Riemann zeta function, and has the same non-trivial zeros. The functional relationship between the two (Aiken, 2019) is:

η(z, 1) = (1 – 2 ^{1 – z} ) ζ (z, 1).

Other functions are closely connected with the Dirichlet eta function, including the Hurwitz-Euler eta function (which is an alternating sign version of the Hurwitz zeta function). Specifically, for x = 0, η (z, x + 1) the Hurwitz-Euler eta function becomes the eta function. The Dirichlet eta function is also a special case of the polylogarithm function.

History and Uses

The Dirichlet eta function was first investigated by Leonhard Euler who used the notation M(s). It is used mainly in number theory, particularly in studying the distribution of prime numbers and as an equivalent for the Riemann Hypothesis.

Numerous authors have used the function to explore the Riemann hypothesis, including Ronald L. Fox, who surmised that:

“Numerical display of the behavior of t strings originating inside the critical strip for the Dirichlet Eta function provides strong visual evidence for why the Riemann hypothesis is most likely true”.

References

Aiken, E. (2019). Modified Commutation Relationships from the Berry-Keating Program.
Bagdasaryan, A. (2010). An Elementary and Real Approach to Values of the Riemann Zeta Function.
Fox, R. (2019). A Conjecture Regarding the Riemann Hypothesis.
Gourdon, X. & Sebah, P. (2004). The Riemann Zeta-function ζ(s) :
generalities. Numbers, constants and computation. Retrieved December 7, 2019 from: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.
Hu, S. & Kim, M. (2015). THE (S, {2})-IWASAWA THEORY.
Hu, S. & Kim, M. (2019). On Dirichlet’s Lambda Function.

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