Dirichlet eta function

E466247

Dirichlet series mathematical function special function

The Dirichlet eta function is an alternating Dirichlet series closely related to the Riemann zeta function and used in analytic number theory, particularly for studying series convergence and analytic continuation.

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All labels observed (3)

Label	Occurrences
Dirichlet eta function canonical	3
Dirichlet lambda function	1
alternating zeta function	1

How this entity was disambiguated

This entity first appeared as the object of triple T4746237 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.

NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07

Target entity: Dirichlet eta function
Context triple: [Peter Gustav Lejeune Dirichlet, notableWork, Dirichlet eta function]

A. Riemann zeta function
The Riemann zeta function is a complex-valued function central to analytic number theory, whose properties—especially the distribution of its zeros—are deeply connected to the distribution of prime numbers.
B. Riemann–Siegel theta function
The Riemann–Siegel theta function is a special function that appears in the study of the Riemann zeta function, used to express its values on the critical line in a form suitable for high-precision numerical computation.
C. Dirichlet series
A Dirichlet series is an infinite series of the form ∑ aₙ/nˢ, fundamental in analytic number theory for studying arithmetic functions and L-functions.
D. Riemann–Siegel formula
The Riemann–Siegel formula is an asymptotic expression that efficiently approximates the Riemann zeta function on the critical line, playing a key role in the numerical study of its zeros.
E. Ramanujan tau function
The Ramanujan tau function is a multiplicative arithmetic function arising from the Fourier coefficients of a modular discriminant form, central to the study of modular forms and number theory.
F. None of above. chosen
G. Unsure - the case is ambiguous/there is not enough information to decide.

NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07

Target entity: Dirichlet eta function
Target entity description: The Dirichlet eta function is an alternating Dirichlet series closely related to the Riemann zeta function and used in analytic number theory, particularly for studying series convergence and analytic continuation.

A. Riemann zeta function
The Riemann zeta function is a complex-valued function central to analytic number theory, whose properties—especially the distribution of its zeros—are deeply connected to the distribution of prime numbers.
B. Riemann–Siegel theta function
The Riemann–Siegel theta function is a special function that appears in the study of the Riemann zeta function, used to express its values on the critical line in a form suitable for high-precision numerical computation.
C. Dirichlet series
A Dirichlet series is an infinite series of the form ∑ aₙ/nˢ, fundamental in analytic number theory for studying arithmetic functions and L-functions.
D. Riemann–Siegel formula
The Riemann–Siegel formula is an asymptotic expression that efficiently approximates the Riemann zeta function on the critical line, playing a key role in the numerical study of its zeros.
E. Ramanujan tau function
The Ramanujan tau function is a multiplicative arithmetic function arising from the Fourier coefficients of a modular discriminant form, central to the study of modular forms and number theory.
F. None of above. chosen

Statements (47)

Predicate	Object
instanceOf	Dirichlet series ⓘ mathematical function ⓘ special function ⓘ
alsoKnownAs	alternating zeta function ⓘ linked to: Dirichlet eta function
analyticContinuation	entire function ⓘ
category	L-function ⓘ
classification	alternating Dirichlet L-series with principal character modulo 2 ⓘ
codomain	complex numbers ⓘ
convergenceProperty	absolutely convergent for Re(s) > 1 ⓘ conditionally convergent for 0 < Re(s) ≤ 1 ⓘ converges for Re(s) > 0 ⓘ
definition	η(s) = ∑_{n=1}^{∞} (-1)^{n-1} / n^s ⓘ
definitionDomain	Re(s) > 0 ⓘ
domain	complex plane ⓘ
EulerTransformation	admits Euler summation acceleration ⓘ
expansionAround	admits Taylor expansion around s = 0 ⓘ
field	analytic number theory ⓘ complex analysis ⓘ
functionalEquationRelation	related to functional equation of Riemann zeta function ⓘ
growthProperty	of finite order as an entire function ⓘ
namedAfter	Peter Gustav Lejeune Dirichlet ⓘ
property	Dirichlet series with real coefficients ⓘ alternating sign coefficients ⓘ entire extension obtained via relation to ζ(s) ⓘ
relatedConcept	Dirichlet beta function ⓘ Dirichlet lambda function ⓘ linked to: Dirichlet eta function Hurwitz zeta function ⓘ
relatedTo	Riemann zeta function ⓘ
relationToBernoulliNumbers	values at negative integers expressible via Bernoulli numbers ⓘ
relationToPolylogarithm	η(s) = (1 - 2^{1-s}) Li_s(1) ⓘ
relationToZeta	η(s) = (1 - 2^{1-s}) ζ(s) ⓘ
seriesRepresentation	η(s) = 1 - 2^{-s} + 3^{-s} - 4^{-s} + ⋯ ⓘ
seriesType	alternating series ⓘ
singularityStructure	no poles in the complex plane ⓘ
symbol	η(s) ⓘ
usedFor	analytic continuation of the Riemann zeta function ⓘ regularization of divergent series ⓘ studying convergence of Dirichlet series ⓘ
usedIn	study of the Riemann hypothesis ⓘ summation of Grandi-type series ⓘ
valueAt	η(0) = 1/2 ⓘ η(1) = ln(2) ⓘ η(2) = π^2 / 12 ⓘ η(−1) = 1/4 ⓘ η(−2n) = 0 for positive integer n ⓘ
zeroType	has trivial zeros at negative even integers ⓘ nontrivial zeros correspond to nontrivial zeros of ζ(s) ⓘ

How these facts were elicited

Referenced by (5)

Full triples — surface form annotated when it differs from this entity's canonical label.

Peter Gustav Lejeune Dirichlet → notableWork → Dirichlet eta function ⓘ

eta → usedAsSymbolFor → Dirichlet eta function ⓘ

subject linked to: Greek letter eta

Dirichlet → knownFor → Dirichlet eta function ⓘ

Dirichlet eta function → alsoKnownAs → alternating zeta function ⓘ

linked to: Dirichlet eta function

Dirichlet eta function → relatedConcept → Dirichlet lambda function ⓘ