DirichletEta

The Dirichlet eta function $\eta(s) = \sum_{n\ge1} (-1)^{n-1} n^{-s} = (1 - 2^{1-s})\,\zeta(s)$, the alternating zeta — entire, with $\eta(1) = \ln 2$.

✓ checked against that system · ! checked, with disagreements (hover a chip)WikipediaDirichlet eta function↗MathWorldDirichletEtaFunction↗WikidataQ973313↗DLMF25.2↗Wolfram LanguageDirichletEta!↗

DirichletEta(s)the Dirichlet eta function $\eta(s)$.

SymPydirichlet_eta(s)mpmathaltzeta(s)Juliaenumeratio_ball(s -> -polylog(s, CC(-1)), s)

Typed signatures

Type	Package	Applies
(complex | infinity) -> number	analyticreplaces compute-engine	always
(complex | infinity) -> number	compute-engine	always

Domain: Special functions

Details

• $\eta(s) = (1 - 2^{1-s})\zeta(s)$: the alternating series converges for $\operatorname{Re}(s) > 0$, and the factor cancels $\zeta$'s pole, so $\eta$ is entire with $\eta(1) = \ln 2$.
• Integer values follow from Zeta's: $\eta(2) = \pi^2/12$, $\eta(0) = \tfrac12$, $\eta(-1) = \tfrac14$, and $\eta(3) = \tfrac34\zeta(3)$ stays in terms of $\zeta(3)$.
• Zeros: the nontrivial zeros of $\zeta$, plus those of $1 - 2^{1-s}$ on the line $\operatorname{Re}(s) = 1$.
• Plain evaluation reduces exact integer $s$ through $\zeta$; other exact $s$ stays symbolic (as in Wolfram) until N() or a floating-point argument. Numerically, complex $s$ is supported; within $0.25$ of $s = 1$ the alternating series is summed directly (Euler transform, via LerchPhi) so no pole is cancelled.

Plot

ComplexPlot

real slice

Examples

Basic #

$\eta(1) = \ln 2$ — the alternating harmonic series

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In$\mathrm{DirichletEta}(1)$Out$\ln(2)$

$\eta(2) = \pi^2/12$

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In$\mathrm{DirichletEta}(2)$Out$\frac{π^2}{12}$

$\eta(3) = \tfrac34\zeta(3)$ — no closed form beyond Apéry's constant

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In$\mathrm{DirichletEta}(3)$Out$\frac{3ζ(3)}{4}$

On the critical line — through $(1 - 2^{1-s})\zeta(s)$, so it carries the engine's working precision

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In$\mathrm{DirichletEta}(0.5)$Out$0.604\,898\,643\,421\,630\,370\,245$

Numeric for a floating-point argument

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In$\mathrm{DirichletEta}(1.5)$Out$0.765\,147\,024\,625\,407\,945\,367$

Scope #

Complex arguments

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In$\mathrm{DirichletEta}(3.5+2.5i)$Out$1.000\,109\,727\,804\,697\,8+0.078\,128\,912\,041\,483\,16i$

Threads elementwise over an array

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In$\mathrm{DirichletEta}(\bigl\lbrack\bigl\lbrack\frac{1}{2}, -1\bigr\rbrack, \bigl\lbrack0, \frac{1}{2}\bigr\rbrack\bigr\rbrack)$Out$\bigl\lbrack\bigl\lbrack\mathrm{DirichletEta}(\frac{1}{2}), \frac{1}{4}\bigr\rbrack, \bigl\lbrack\frac{1}{2}, \mathrm{DirichletEta}(\frac{1}{2})\bigr\rbrack\bigr\rbrack$

$\eta(s) \to 1$ as $s \to \infty$

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In$\mathrm{DirichletEta}(∞)$Out$1$

Interval arithmetic over an increasing stretch

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In$\mathrm{DirichletEta}(\mathrm{Interval}(1.5, 1.6))$Out$\mathrm{Interval}(0.765\,147\,024\,625\,407\,8, 0.777\,722\,726\,661\,128\,8)$

Uncertainty propagation: $\eta(2 \pm 0.01) = 0.82247 \pm 0.00101$

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In$\mathrm{DirichletEta}(\mathrm{Around}(2, 0.01))$Out$\mathrm{Around}(0.822\,467\,033\,424\,113\,2, 0.001\,013\,165\,781\,635\,045)$

To 30 significant digits, the last one correctly rounded

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In$N(\mathrm{DirichletEta}(\frac{1}{2}), 30)$Out$0.604\,898\,643\,421\,630\,370\,247\,265\,914\,236$

Properties #

Nonpositive integers: $\eta(-n) = (2^{n+1} - 1)\,B_{n+1}/(n+1)$, rational

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In$\mathrm{DirichletEta}(\bigl\lbrack0, -1, -2, -3\bigr\rbrack)$Out$\bigl\lbrack\frac{1}{2}, \frac{1}{4}, 0, -\frac{1}{8}\bigr\rbrack$

$\eta(4) = \tfrac78\zeta(4) = 7\pi^4/720$

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In$\mathrm{DirichletEta}(4)$Out$\frac{7π^4}{720}$

Rewriting in terms of Zeta with FunctionExpand: $\eta(s) = (1 - 2^{1-s})\zeta(s)$

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In$\mathrm{FunctionExpand}(\mathrm{DirichletEta}(s))$Out$(1-2^{1-s})ζ(s)$

Possible issues #

An exact non-integer argument stays symbolic until N()

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In$\mathrm{DirichletEta}(\frac{1}{2})$Out$\mathrm{DirichletEta}(\frac{1}{2})$

About

Examples	15
Checked against	Wolfram Language 11 agree4 differSymPy 12 agree1 differmpmath 12 agreeJulia 11 agree1 error
Crosswalk	WikipediaMathWorldWikidataDLMFWolfram LanguageSymPympmathJulia

Implementation

referenceenginenotatio

The defining identity — except at s = 1, where the native kernel sums the alternating series instead.

nativeenginetypescriptpackages/ce-patches/src/compute-engine/numerics/dirichlet.ts

mappedexternalwolfram / mpmathpackages/oracle/src/mappings.ts

DirichletEta[s]; mpmath.altzeta(s).

mappedsympypackages/oracle/src/mappings.ts

mappedmpmathpackages/oracle/src/mappings.ts

mappedjuliapackages/oracle/src/mappings.ts

See also: Zeta, DirichletBeta, LerchPhi, PolyLog