Dirichlet eta function
E466247
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.
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
Dirichlet series
ⓘ
mathematical function ⓘ special function ⓘ |
| alsoKnownAs | alternating zeta function NERFINISHED ⓘ |
| 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 NERFINISHED ⓘ |
| property |
Dirichlet series with real coefficients
ⓘ
alternating sign coefficients ⓘ entire extension obtained via relation to ζ(s) ⓘ |
| relatedConcept |
Dirichlet beta function
NERFINISHED
ⓘ
Dirichlet lambda function NERFINISHED ⓘ Hurwitz zeta function NERFINISHED ⓘ |
| relatedTo | Riemann zeta function NERFINISHED ⓘ |
| 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) ⓘ |
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
subject surface form:
eta