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Hurwitz zeta function

Table of contents: Definitions - Illustrations - Domain and range - Specific values - Series representations - Integral representations - Functional equations - Derivatives and differential equations - Euler-Maclaurin formula - Representation of other functions

Definitions

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Symbol: HurwitzZeta ζ ⁣(s,a)\zeta\!\left(s, a\right) Hurwitz zeta function

Illustrations

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Image: Plot of ζ ⁣(s,a)\zeta\!\left(s, a\right) on s[25,11]s \in \left[-25, 11\right] for a{0.6,0.8,1.4}a \in \left\{0.6, 0.8, 1.4\right\}
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Image: X-ray of ζ ⁣(s,1+i2)\zeta\!\left(s, 1 + \frac{i}{2}\right) on s[20,20]+[20,20]is \in \left[-20, 20\right] + \left[-20, 20\right] i
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Image: X-ray of ζ ⁣(2+3i,a)\zeta\!\left(2 + 3 i, a\right) on a[5,5]+[5,5]ia \in \left[-5, 5\right] + \left[-5, 5\right] i

Domain and range

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(sC{1}andaC{0,1,})    (ζ ⁣(s,a)C)\left(s \in \mathbb{C} \setminus \left\{1\right\} \,\mathbin{\operatorname{and}}\, a \in \mathbb{C} \setminus \{0, -1, \ldots\}\right) \implies \left(\zeta\!\left(s, a\right) \in \mathbb{C}\right)
ad269f
(sCandRe(s)<0andaC)    (ζ ⁣(s,a)C)\left(s \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \operatorname{Re}(s) < 0 \,\mathbin{\operatorname{and}}\, a \in \mathbb{C}\right) \implies \left(\zeta\!\left(s, a\right) \in \mathbb{C}\right)
e7224b
(sR{1}anda(0,))    (ζ ⁣(s,a)R)\left(s \in \mathbb{R} \setminus \left\{1\right\} \,\mathbin{\operatorname{and}}\, a \in \left(0, \infty\right)\right) \implies \left(\zeta\!\left(s, a\right) \in \mathbb{R}\right)
b0f500
(sZ{1}andaR{0,1,})    (ζ ⁣(s,a)R)\left(s \in \mathbb{Z} \setminus \left\{1\right\} \,\mathbin{\operatorname{and}}\, a \in \mathbb{R} \setminus \{0, -1, \ldots\}\right) \implies \left(\zeta\!\left(s, a\right) \in \mathbb{R}\right)
cc523f
(s{0,1,}andaQ)    (ζ ⁣(s,a)Q)\left(s \in \{0, -1, \ldots\} \,\mathbin{\operatorname{and}}\, a \in \mathbb{Q}\right) \implies \left(\zeta\!\left(s, a\right) \in \mathbb{Q}\right)
ec2dd5
(s{0,1,}andaR)    (ζ ⁣(s,a)R)\left(s \in \{0, -1, \ldots\} \,\mathbin{\operatorname{and}}\, a \in \mathbb{R}\right) \implies \left(\zeta\!\left(s, a\right) \in \mathbb{R}\right)
a5980a
(s{0,1,}andaC)    (ζ ⁣(s,a)C)\left(s \in \{0, -1, \ldots\} \,\mathbin{\operatorname{and}}\, a \in \mathbb{C}\right) \implies \left(\zeta\!\left(s, a\right) \in \mathbb{C}\right)
d0b234
(s{1}andaC{0,1,})    (ζ ⁣(s,a){~})\left(s \in \left\{1\right\} \,\mathbin{\operatorname{and}}\, a \in \mathbb{C} \setminus \{0, -1, \ldots\}\right) \implies \left(\zeta\!\left(s, a\right) \in \left\{{\tilde \infty}\right\}\right)
c5d844
(sZ2anda{0,1,})    (ζ ⁣(s,a){~})\left(s \in \mathbb{Z}_{\ge 2} \,\mathbin{\operatorname{and}}\, a \in \{0, -1, \ldots\}\right) \implies \left(\zeta\!\left(s, a\right) \in \left\{{\tilde \infty}\right\}\right)

As a function of the argument

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(aC{0,1,})    (ζ ⁣(s,a) is holomorphic on sC{1})\left(a \in \mathbb{C} \setminus \{0, -1, \ldots\}\right) \implies \left(\zeta\!\left(s, a\right) \text{ is holomorphic on } s \in \mathbb{C} \setminus \left\{1\right\}\right)
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(aC{0,1,})    (ζ ⁣(s,a) is meromorphic on sC)\left(a \in \mathbb{C} \setminus \{0, -1, \ldots\}\right) \implies \left(\zeta\!\left(s, a\right) \text{ is meromorphic on } s \in \mathbb{C}\right)
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(aC)    (ζ ⁣(s,a) is holomorphic on s{t:tCandRe(t)<0})\left(a \in \mathbb{C}\right) \implies \left(\zeta\!\left(s, a\right) \text{ is holomorphic on } s \in \left\{ t : t \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \operatorname{Re}(t) < 0 \right\}\right)

As a function of the parameter

93e149
(sC{1})    (ζ ⁣(s,a) is holomorphic on aC(,0])\left(s \in \mathbb{C} \setminus \left\{1\right\}\right) \implies \left(\zeta\!\left(s, a\right) \text{ is holomorphic on } a \in \mathbb{C} \setminus \left(-\infty, 0\right]\right)
8c7cdb
(sZ2)    (ζ ⁣(s,a) is holomorphic on aC{0,1,})\left(s \in \mathbb{Z}_{\ge 2}\right) \implies \left(\zeta\!\left(s, a\right) \text{ is holomorphic on } a \in \mathbb{C} \setminus \{0, -1, \ldots\}\right)
05c2dd
(sZ2)    (ζ ⁣(s,a) is meromorphic on aC)\left(s \in \mathbb{Z}_{\ge 2}\right) \implies \left(\zeta\!\left(s, a\right) \text{ is meromorphic on } a \in \mathbb{C}\right)
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(s{0,1,})    (ζ ⁣(s,a) is holomorphic on aC)\left(s \in \{0, -1, \ldots\}\right) \implies \left(\zeta\!\left(s, a\right) \text{ is holomorphic on } a \in \mathbb{C}\right)

Specific values

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ζ ⁣(s,1)=ζ(s)\zeta\!\left(s, 1\right) = \zeta(s)
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ζ ⁣(s,2)=ζ(s)1\zeta\!\left(s, 2\right) = \zeta(s) - 1
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ζ ⁣(s,3)=ζ(s)112s\zeta\!\left(s, 3\right) = \zeta(s) - 1 - \frac{1}{{2}^{s}}
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ζ ⁣(s,n)=ζ(s)k=1n11ks\zeta\!\left(s, n\right) = \zeta(s) - \sum_{k=1}^{n - 1} \frac{1}{{k}^{s}}
af7d3d
ζ ⁣(s,12)=(2s1)ζ(s)\zeta\!\left(s, \frac{1}{2}\right) = \left({2}^{s} - 1\right) \zeta(s)
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ζ ⁣(s,32)=(2s1)ζ(s)2s\zeta\!\left(s, \frac{3}{2}\right) = \left({2}^{s} - 1\right) \zeta(s) - {2}^{s}
6c3523
ζ ⁣(s,12+n)=(2s1)ζ(s)2sk=0n11(2k+1)s\zeta\!\left(s, \frac{1}{2} + n\right) = \left({2}^{s} - 1\right) \zeta(s) - {2}^{s} \sum_{k=0}^{n - 1} \frac{1}{{\left(2 k + 1\right)}^{s}}
8bbb6f
ζ ⁣(s,14)+ζ ⁣(s,34)=2s(2s1)ζ(s)\zeta\!\left(s, \frac{1}{4}\right) + \zeta\!\left(s, \frac{3}{4}\right) = {2}^{s} \left({2}^{s} - 1\right) \zeta(s)
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ζ ⁣(s,16)+ζ ⁣(s,56)=(2s1)(3s1)ζ(s)\zeta\!\left(s, \frac{1}{6}\right) + \zeta\!\left(s, \frac{5}{6}\right) = \left({2}^{s} - 1\right) \left({3}^{s} - 1\right) \zeta(s)
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ζ ⁣(2,1)=π26\zeta\!\left(2, 1\right) = \frac{{\pi}^{2}}{6}
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ζ ⁣(2,2)=π261\zeta\!\left(2, 2\right) = \frac{{\pi}^{2}}{6} - 1
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ζ ⁣(3,1)=ζ(3)\zeta\!\left(3, 1\right) = \zeta(3)
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ζ ⁣(3,2)=ζ(3)1\zeta\!\left(3, 2\right) = \zeta(3) - 1
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ζ ⁣(4,1)=π490\zeta\!\left(4, 1\right) = \frac{{\pi}^{4}}{90}
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ζ ⁣(4,2)=π4901\zeta\!\left(4, 2\right) = \frac{{\pi}^{4}}{90} - 1
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ζ ⁣(2,12)=π22\zeta\!\left(2, \frac{1}{2}\right) = \frac{{\pi}^{2}}{2}
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ζ ⁣(3,12)=7ζ(3)\zeta\!\left(3, \frac{1}{2}\right) = 7 \zeta(3)
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ζ ⁣(4,12)=π46\zeta\!\left(4, \frac{1}{2}\right) = \frac{{\pi}^{4}}{6}
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ζ ⁣(2,14)=π2+8G\zeta\!\left(2, \frac{1}{4}\right) = {\pi}^{2} + 8 G
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ζ ⁣(2,34)=π28G\zeta\!\left(2, \frac{3}{4}\right) = {\pi}^{2} - 8 G
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ζ ⁣(3,14)=28ζ(3)+π3\zeta\!\left(3, \frac{1}{4}\right) = 28 \zeta(3) + {\pi}^{3}
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ζ ⁣(3,34)=28ζ(3)π3\zeta\!\left(3, \frac{3}{4}\right) = 28 \zeta(3) - {\pi}^{3}
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ζ ⁣(3,16)=91ζ(3)+23π3\zeta\!\left(3, \frac{1}{6}\right) = 91 \zeta(3) + 2 \sqrt{3} {\pi}^{3}
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ζ ⁣(3,56)=91ζ(3)23π3\zeta\!\left(3, \frac{5}{6}\right) = 91 \zeta(3) - 2 \sqrt{3} {\pi}^{3}
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ζ ⁣(n,a)=(1)n(n1)!ψ(n1) ⁣(a)\zeta\!\left(n, a\right) = \frac{{\left(-1\right)}^{n}}{\left(n - 1\right)!} \psi^{(n - 1)}\!\left(a\right)
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ζ ⁣(1,a)=~\zeta\!\left(1, a\right) = {\tilde \infty}
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ζ ⁣(n,a)=Bn+1 ⁣(a)n+1\zeta\!\left(-n, a\right) = -\frac{B_{n + 1}\!\left(a\right)}{n + 1}
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ζ ⁣(n,0)=Bn+1n+1\zeta\!\left(-n, 0\right) = -\frac{B_{n + 1}}{n + 1}
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ζ ⁣(0,a)=12a\zeta\!\left(0, a\right) = \frac{1}{2} - a
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ζ ⁣(0,0)=12\zeta\!\left(0, 0\right) = \frac{1}{2}
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ζ ⁣(0,12)=0\zeta\!\left(0, \frac{1}{2}\right) = 0
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ζ ⁣(2,a)=a23F2 ⁣(1,a,a,a+1,a+1,1)\zeta\!\left(2, a\right) = {a}^{-2} \,{}_3F_2\!\left(1, a, a, a + 1, a + 1, 1\right)

Series representations

Dirichlet series

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ζ ⁣(s,a)=n=01(n+a)s\zeta\!\left(s, a\right) = \sum_{n=0}^{\infty} \frac{1}{{\left(n + a\right)}^{s}}
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ζ(r) ⁣(s,a)=(1)rn=0logr ⁣(n+a)(n+a)s\zeta^{(r)}\!\left(s, a\right) = {\left(-1\right)}^{r} \sum_{n=0}^{\infty} \frac{\log^{r}\!\left(n + a\right)}{{\left(n + a\right)}^{s}}
0bd6aa
ζ ⁣(s,N)=n=N1ns\zeta\!\left(s, N\right) = \sum_{n=N}^{\infty} \frac{1}{{n}^{s}}

Laurent series

Related topic: Stieltjes constants
60c6da
ζ ⁣(s,a)=1s1+n=0(1)nn!γn ⁣(a)(s1)n\zeta\!\left(s, a\right) = \frac{1}{s - 1} + \sum_{n=0}^{\infty} \frac{{\left(-1\right)}^{n}}{n !} \gamma_{n}\!\left(a\right) {\left(s - 1\right)}^{n}

Integral representations

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ζ ⁣(s,a)=π2(s1)(a12+ix)1scosh2 ⁣(πx)dx\zeta\!\left(s, a\right) = \frac{\pi}{2 \left(s - 1\right)} \int_{-\infty}^{\infty} \frac{{\left(a - \frac{1}{2} + i x\right)}^{1 - s}}{\cosh^{2}\!\left(\pi x\right)} \, dx
498036
ζ ⁣(s,a)=1Γ(s)0xs1eax1exdx\zeta\!\left(s, a\right) = \frac{1}{\Gamma(s)} \int_{0}^{\infty} \frac{{x}^{s - 1} {e}^{-a x}}{1 - {e}^{-x}} \, dx

Functional equations

Recurrence relations

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ζ ⁣(s,a+1)=ζ ⁣(s,a)1as\zeta\!\left(s, a + 1\right) = \zeta\!\left(s, a\right) - \frac{1}{{a}^{s}}
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ζ ⁣(s,a+N)=ζ ⁣(s,a)n=0N11(n+a)s\zeta\!\left(s, a + N\right) = \zeta\!\left(s, a\right) - \sum_{n=0}^{N - 1} \frac{1}{{\left(n + a\right)}^{s}}
95e270
ζ(r) ⁣(s,a+N)=ζ(r) ⁣(s,a)+(1)r+1k=0N1logr ⁣(a+k)(a+k)s\zeta^{(r)}\!\left(s, a + N\right) = \zeta^{(r)}\!\left(s, a\right) + {\left(-1\right)}^{r + 1} \sum_{k=0}^{N - 1} \frac{\log^{r}\!\left(a + k\right)}{{\left(a + k\right)}^{s}}

Multiplication formula

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ζ ⁣(s,Na)=1Nsk=0N1ζ ⁣(s,a+kN)\zeta\!\left(s, N a\right) = \frac{1}{{N}^{s}} \sum_{k=0}^{N - 1} \zeta\!\left(s, a + \frac{k}{N}\right)
ebc49c
ζ ⁣(s,a)=12s(ζ ⁣(s,a2)+ζ ⁣(s,a+12))\zeta\!\left(s, a\right) = \frac{1}{{2}^{s}} \left(\zeta\!\left(s, \frac{a}{2}\right) + \zeta\!\left(s, \frac{a + 1}{2}\right)\right)
7d9feb
ζ ⁣(s,a)=1Nsk=0N1ζ ⁣(s,a+kN)\zeta\!\left(s, a\right) = \frac{1}{{N}^{s}} \sum_{k=0}^{N - 1} \zeta\!\left(s, \frac{a + k}{N}\right)

Reflection formula

69a1a9
ζ ⁣(1s,pq)=2Γ(s)(2πq)sk=1qcos ⁣(πs22πkpq)ζ ⁣(s,kq)\zeta\!\left(1 - s, \frac{p}{q}\right) = \frac{2 \Gamma(s)}{{\left(2 \pi q\right)}^{s}} \sum_{k=1}^{q} \cos\!\left(\frac{\pi s}{2} - \frac{2 \pi k p}{q}\right) \zeta\!\left(s, \frac{k}{q}\right)

Derivatives and differential equations

Argument derivatives

3ba544
ddsζ ⁣(s,a)=ζ ⁣(s,a)\frac{d}{d s}\, \zeta\!\left(s, a\right) = \zeta'\!\left(s, a\right)
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drdsrζ ⁣(s,a)=ζ(r) ⁣(s,a)\frac{d^{r}}{{d s}^{r}} \zeta\!\left(s, a\right) = \zeta^{(r)}\!\left(s, a\right)

Parameter derivatives

83065e
ddaζ ⁣(s,a)=sζ ⁣(s+1,a)\frac{d}{d a}\, \zeta\!\left(s, a\right) = -s \zeta\!\left(s + 1, a\right)
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drdarζ ⁣(s,a)=(1sr)rζ ⁣(s+r,a)\frac{d^{r}}{{d a}^{r}} \zeta\!\left(s, a\right) = \left(1 - s - r\right)_{r} \zeta\!\left(s + r, a\right)

Euler-Maclaurin formula

d25d10
ζ ⁣(s,a)=k=0N11(a+k)s+(a+N)1ss1+1(a+N)s(12+k=1MB2k(2k)!(s)2k1(a+N)2k1)NB2M ⁣(tt)(2M)!(s)2M(a+t)s+2Mdt\zeta\!\left(s, a\right) = \sum_{k=0}^{N - 1} \frac{1}{{\left(a + k\right)}^{s}} + \frac{{\left(a + N\right)}^{1 - s}}{s - 1} + \frac{1}{{\left(a + N\right)}^{s}} \left(\frac{1}{2} + \sum_{k=1}^{M} \frac{B_{2 k}}{\left(2 k\right)!} \frac{\left(s\right)_{2 k - 1}}{{\left(a + N\right)}^{2 k - 1}}\right) - \int_{N}^{\infty} \frac{B_{2 M}\!\left(t - \left\lfloor t \right\rfloor\right)}{\left(2 M\right)!} \frac{\left(s\right)_{2 M}}{{\left(a + t\right)}^{s + 2 M}} \, dt

Representation of other functions

Riemann zeta function

Related topic: Riemann zeta function
febdd2
ζ(s)=ζ ⁣(s,1)\zeta(s) = \zeta\!\left(s, 1\right)

Bernoulli polynomials

Related topic: Bernoulli numbers and polynomials
4228cd
Bn ⁣(z)=nζ ⁣(1n,z)B_{n}\!\left(z\right) = -n \zeta\!\left(1 - n, z\right)

Gamma and related functions

Related topics: Digamma function, Barnes G-function
53026a
Γ(z)=2πexp ⁣(ζ ⁣(0,z))\Gamma(z) = \sqrt{2 \pi} \exp\!\left(\zeta'\!\left(0, z\right)\right)
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logΓ(z)=ζ ⁣(0,z)+log ⁣(2π)2\log \Gamma(z) = \zeta'\!\left(0, z\right) + \frac{\log\!\left(2 \pi\right)}{2}
693e0e
ψ ⁣(z)=lims1[1s1ζ ⁣(s,z)]\psi\!\left(z\right) = \lim_{s \to 1} \left[\frac{1}{s - 1} - \zeta\!\left(s, z\right)\right]
bba4ec
ψ(m) ⁣(z)=(1)m+1m!ζ ⁣(m+1,z)\psi^{(m)}\!\left(z\right) = {\left(-1\right)}^{m + 1} m ! \zeta\!\left(m + 1, z\right)
e05807
logG(z)=(z1)logΓ(z)ζ ⁣(1,z)+ζ(1)\log G(z) = \left(z - 1\right) \log \Gamma(z) - \zeta'\!\left(-1, z\right) + \zeta'(-1)

Dirichlet L-functions

Related topic: Dirichlet L-functions
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L ⁣(s,χ)=1qsk=1qχ(k)ζ ⁣(s,kq)L\!\left(s, \chi\right) = \frac{1}{{q}^{s}} \sum_{k=1}^{q} \chi(k) \zeta\!\left(s, \frac{k}{q}\right)
4c3678
ζ ⁣(s,kq)=qsφ(q)χGqχ(k)L ⁣(s,χ)\zeta\!\left(s, \frac{k}{q}\right) = \frac{{q}^{s}}{\varphi(q)} \sum_{\chi \in G_{q}} \overline{\chi(k)} L\!\left(s, \chi\right)

Polylogarithms

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Lis ⁣(z)=Γ ⁣(1s)(2π)1s(i1sζ ⁣(s,12+log ⁣(z)2πi)+is1ζ ⁣(1s,12log ⁣(z)2πi))\operatorname{Li}_{s}\!\left(z\right) = \frac{\Gamma\!\left(1 - s\right)}{{\left(2 \pi\right)}^{1 - s}} \left({i}^{1 - s} \zeta\!\left(s, \frac{1}{2} + \frac{\log\!\left(-z\right)}{2 \pi i}\right) + {i}^{s - 1} \zeta\!\left(1 - s, \frac{1}{2} - \frac{\log\!\left(-z\right)}{2 \pi i}\right)\right)

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2019-11-11 15:50:15.016492 UTC