Dirichlet eta function

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In mathematics, in the area of analytic number theory, the Dirichlet eta function is defined by the following Dirichlet series, which converges for any complex number having real part > 0: $${\displaystyle \eta (s)=\sum _{n=1}^{\infty }{(-1)^{n-1} \over n^{s}}={\frac {1}{1^{s}}}-{\frac {1}{2^{s}}}+{\frac {1}{3^{s}}}-{\frac {1}{4^{s}}}+\cdots .}$$

This Dirichlet series is the alternating sum corresponding to the Dirichlet series expansion of the Riemann zeta function, ζ(s) — and for this reason the Dirichlet eta function is also known as the alternating zeta function, also denoted ζ*(s). The following relation holds: $${\displaystyle \eta (s)=\left(1-2^{1-s}\right)\zeta (s)}$$

Both the Dirichlet eta function and the Riemann zeta function are special cases of polylogarithms.

While the Dirichlet series expansion for the eta function is convergent only for any complex number s with real part > 0, it is Abel summable for any complex number. This serves to define the eta function as an entire function. (The above relation and the facts that the eta function is entire and ${\displaystyle \eta (1)\neq 0}$ together show the zeta function is meromorphic with a simple pole at s = 1, and possibly additional poles at the other zeros of the factor ${\displaystyle 1-2^{1-s}}$, although in fact these hypothetical additional poles do not exist.)

Equivalently, we may begin by defining $${\displaystyle \eta (s)={\frac {1}{\Gamma (s)}}\int _{0}^{\infty }{\frac {x^{s-1}}{e^{x}+1}}{dx}}$$ which is also defined in the region of positive real part (${\displaystyle \Gamma (s)}$ represents the gamma function). This gives the eta function as a Mellin transform.

Hardy gave a simple proof of the functional equation for the eta function,^[2] which is $${\displaystyle \eta (-s)=2{\frac {1-2^{-s-1}}{1-2^{-s}}}\pi ^{-s-1}s\sin \left({\pi s \over 2}\right)\Gamma (s)\eta (s+1).}$$

From this, one immediately has the functional equation of the zeta function also, as well as another means to extend the definition of eta to the entire complex plane.