Hurwitz zeta function
E931273
Dirichlet series
complex analytic function
generalization of Riemann zeta function
meromorphic function
special function
The Hurwitz zeta function is a complex analytic function that generalizes the Riemann zeta function by introducing a shift parameter, playing a key role in analytic number theory and special function theory.
Statements (49)
| Predicate | Object |
|---|---|
| instanceOf |
Dirichlet series
ⓘ
complex analytic function ⓘ generalization of Riemann zeta function ⓘ meromorphic function ⓘ special function ⓘ |
| analyticContinuationDomain | C × (C \ Z_{≤0}) except s = 1 ⓘ |
| appearsIn |
analytic number theory
ⓘ
quantum field theory ⓘ spectral theory ⓘ statistical mechanics ⓘ |
| coefficientInLaurentExpansion | Stieltjes constants generalized by parameter a ⓘ |
| convergesFor | Re(s) > 1 ⓘ |
| definingConditionOnParameter | Re(a) > 0 ⓘ |
| definition | ζ(s,a) = Σ_{n=0}^{∞} (n + a)^{-s} for Re(s) > 1 and Re(a) > 0 ⓘ |
| differentiationRelation | ∂/∂a ζ(s,a) = -s ζ(s+1,a) ⓘ |
| domainOfParameter | complex numbers ⓘ |
| domainOfVariable | complex numbers ⓘ |
| fieldOfStudy |
complex analysis
ⓘ
number theory ⓘ special function theory ⓘ |
| functionalEquationType | generalized functional equation extending that of Riemann zeta function ⓘ |
| generalizes |
Hurwitz–Lerch zeta function (as a special case with z = 1)
NERFINISHED
ⓘ
Riemann zeta function NERFINISHED ⓘ |
| growthProperty | of polynomial growth in vertical strips away from s = 1 ⓘ |
| hasAnalyticContinuation | yes ⓘ |
| hasLaurentExpansionAt | s = 1 ⓘ |
| hasSimplePoleAt | s = 1 ⓘ |
| namedAfter | Adolf Hurwitz NERFINISHED ⓘ |
| parameter | a ⓘ |
| relatedTo |
Bernoulli numbers
NERFINISHED
ⓘ
Bernoulli polynomials NERFINISHED ⓘ Dirichlet L-functions NERFINISHED ⓘ Gamma function NERFINISHED ⓘ Riemann zeta function NERFINISHED ⓘ polygamma functions ⓘ polylogarithm ⓘ |
| residueAt | (s = 1, residue = 1) ⓘ |
| seriesType | Dirichlet series in (n+a)^{-s} ⓘ |
| specialCase |
ζ(s,1) = Riemann zeta function ζ(s)
ⓘ
ζ(s,1/2) related to Dirichlet L-function L(s,χ_2) ⓘ ζ(s,a) with rational a expressible via Dirichlet L-functions ⓘ |
| symbol | ζ(s,a) ⓘ |
| usedFor |
evaluation of series
ⓘ
regularization of divergent sums ⓘ special values at integers ⓘ study of distribution of arithmetic sequences ⓘ |
| valueAt |
ζ(-n,a) = -B_{n+1}(a)/(n+1) for n ∈ N
ⓘ
ζ(0,a) = 1/2 - a ⓘ |
| variable | s ⓘ |
Referenced by (3)
Full triples — surface form annotated when it differs from this entity's canonical label.
subject surface form:
Adolf Hurwitz