Editing Dirichlet eta function (section)

{{Short description|Function in analytic number theory}}{{For|the modular form|Dedekind eta function}}
{{More footnotes|date=August 2017}}

[[File:Dirichlet eta function.png|right|thumb|Color representation of the Dirichlet eta function.  It is generated as a [[Matplotlib]] plot using a version of the [[Domain coloring]] method.<ref>{{Cite web|url=http://nbviewer.ipython.org/github/empet/Math/blob/master/DomainColoring.ipynb | title=Jupyter Notebook Viewer}}</ref>]]

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:
<math display="block">\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.</math>

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:
<math display="block">\eta(s) = \left(1-2^{1-s}\right) \zeta(s)</math>

Both the Dirichlet eta function and the Riemann zeta function are special cases of [[Polylogarithm#Relationship to other functions|polylogarithms]].

While the Dirichlet series expansion for the eta function is convergent only for any [[complex number]] ''s'' with real part > 0, it is [[divergent series|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 <math>\eta(1) \neq 0</math> together show the zeta function is [[meromorphic function|meromorphic]] with a simple [[pole (complex analysis)|pole]] at ''s'' = 1, and possibly additional poles at the other zeros of the factor <math>1-2^{1-s}</math>, although in fact these hypothetical additional poles do not exist.)

Equivalently, we may begin by defining
<math display="block">\eta(s) = \frac{1}{\Gamma(s)}\int_0^\infty \frac{x^{s-1}}{e^x+1}{dx}</math>
which is also defined in the region of positive real part (<math>\Gamma(s)</math> represents the [[gamma function]]). This gives the eta function as a [[Mellin transform]].

[[G. H. Hardy|Hardy]] gave a simple proof of the [[functional equation]] for the eta function,<ref>Hardy, G. H. (1922). A new proof of the functional equation for the Zeta-function. Matematisk Tidsskrift. B, 71–73. http://www.jstor.org/stable/24529536</ref> which is
<math display="block">\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).</math>

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.