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DirichletEta[s]

gives the Dirichlet eta function TemplateBox[{s}, DirichletEta].

Details
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Examples  
Basic Examples  
Scope  
Properties & Relations  
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DirichletEta

DirichletEta[s]

gives the Dirichlet eta function TemplateBox[{s}, DirichletEta].

Details

  • The Dirichlet eta function is also known as the alternating zeta function.
  • DirichletEta is a mathematical function, suitable for both symbolic and numeric manipulation.
  • For , the Dirichlet eta function is defined as TemplateBox[{s}, DirichletEta]=sum_(n=0)^infty((-1)^n)/((n+1)^s).
  • For certain special arguments, DirichletEta automatically evaluates to exact values.
  • DirichletEta is an entire function with branch cut discontinuities.
  • DirichletEta can be evaluated to arbitrary numerical precision.
  • DirichletEta automatically threads over lists.
  • DirichletEta can be used with Interval and CenteredInterval objects. »

Examples

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Basic Examples  (3)

Plot over the reals:

Wolfram Language code: Plot[DirichletEta[s], {s, -4, 4}]

Visualize in the complex plane:

Wolfram Language code: ComplexPlot3D[DirichletEta[z], {z, 5}]

Compute a special value:

Wolfram Language code: DirichletEta /@ {0, 1, 2}

Scope  (6)

DirichletEta is neither non-decreasing nor non-increasing:

Wolfram Language code: FunctionMonotonicity[DirichletEta[x], x]

DirichletEta is not injective:

Wolfram Language code: FunctionInjective[DirichletEta[x], x]
Wolfram Language code: Plot[{DirichletEta[x], -.05}, {x, -5, 5}]

DirichletEta is neither non-negative nor non-positive:

Wolfram Language code: FunctionSign[DirichletEta[x], x]

DirichletEta is neither convex nor concave:

Wolfram Language code: FunctionConvexity[DirichletEta[x], x]

Compute worst-case guaranteed intervals using Interval and CenteredInterval objects:

Wolfram Language code: DirichletEta[Interval[{1.5, 1.6 }]]
Wolfram Language code: DirichletEta[CenteredInterval[3 + 2I, (1 + I) / 100]]

Or compute average-case statistical intervals using Around:

Wolfram Language code: DirichletEta[ Around[2, 0.01]]

Compute the elementwise values of an array:

Wolfram Language code: DirichletEta[{{1 / 2, -1}, {0, 1 / 2}}]

Or compute the matrix DirichletEta function using MatrixFunction:

Wolfram Language code: MatrixFunction[DirichletEta, {{1 / 2, -1}, {0, 1 / 2}}]//FullSimplify

Properties & Relations  (1)

Verify the interrelationship among the DirichletEta, DirichletLambda and Zeta functions:

Wolfram Language code: (Zeta[s]/2^s) == (DirichletLambda[s]/2^s - 1) == (DirichletEta[s]/2^s - 2)//FullSimplify
Wolfram Research (2014), DirichletEta, Wolfram Language function, https://reference.wolfram.com/language/ref/DirichletEta.html (updated 2022).

Text

Wolfram Research (2014), DirichletEta, Wolfram Language function, https://reference.wolfram.com/language/ref/DirichletEta.html (updated 2022).

CMS

Wolfram Language. 2014. "DirichletEta." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2022. https://reference.wolfram.com/language/ref/DirichletEta.html.

APA

Wolfram Language. (2014). DirichletEta. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/DirichletEta.html

BibTeX

@misc{reference.wolfram_2026_dirichleteta, author="Wolfram Research", title="{DirichletEta}", year="2022", howpublished="\url{https://reference.wolfram.com/language/ref/DirichletEta.html}", note=[Accessed: 12-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_dirichleteta, organization={Wolfram Research}, title={DirichletEta}, year={2022}, url={https://reference.wolfram.com/language/ref/DirichletEta.html}, note=[Accessed: 12-June-2026]}