# Hurwitz zeta function

## Definitions

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

## Illustrations

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

## Domain and range

$\left(s \in \mathbb{C} \setminus \left\{1\right\} \;\mathbin{\operatorname{and}}\; a \in \mathbb{C} \setminus \{0, -1, \ldots\}\right) \;\implies\; \zeta\!\left(s, a\right) \in \mathbb{C}$
$\left(s \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(s) < 0 \;\mathbin{\operatorname{and}}\; a \in \mathbb{C}\right) \;\implies\; \zeta\!\left(s, a\right) \in \mathbb{C}$
$\left(s \in \mathbb{R} \setminus \left\{1\right\} \;\mathbin{\operatorname{and}}\; a \in \left(0, \infty\right)\right) \;\implies\; \zeta\!\left(s, a\right) \in \mathbb{R}$
$\left(s \in \mathbb{Z} \setminus \left\{1\right\} \;\mathbin{\operatorname{and}}\; a \in \mathbb{R} \setminus \{0, -1, \ldots\}\right) \;\implies\; \zeta\!\left(s, a\right) \in \mathbb{R}$
$\left(s \in \{0, -1, \ldots\} \;\mathbin{\operatorname{and}}\; a \in \mathbb{Q}\right) \;\implies\; \zeta\!\left(s, a\right) \in \mathbb{Q}$
$\left(s \in \{0, -1, \ldots\} \;\mathbin{\operatorname{and}}\; a \in \mathbb{R}\right) \;\implies\; \zeta\!\left(s, a\right) \in \mathbb{R}$
$\left(s \in \{0, -1, \ldots\} \;\mathbin{\operatorname{and}}\; a \in \mathbb{C}\right) \;\implies\; \zeta\!\left(s, a\right) \in \mathbb{C}$
$\left(s \in \left\{1\right\} \;\mathbin{\operatorname{and}}\; a \in \mathbb{C} \setminus \{0, -1, \ldots\}\right) \;\implies\; \zeta\!\left(s, a\right) \in \left\{{\tilde \infty}\right\}$
$\left(s \in \mathbb{Z}_{\ge 2} \;\mathbin{\operatorname{and}}\; a \in \{0, -1, \ldots\}\right) \;\implies\; \zeta\!\left(s, a\right) \in \left\{{\tilde \infty}\right\}$

### As a function of the argument

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

### As a function of the parameter

$s \in \mathbb{C} \setminus \left\{1\right\} \;\implies\; \left(\zeta\!\left(s, a\right) \text{ is holomorphic on } a \in \mathbb{C} \setminus \left(-\infty, 0\right]\right)$
$s \in \mathbb{Z}_{\ge 2} \;\implies\; \left(\zeta\!\left(s, a\right) \text{ is holomorphic on } a \in \mathbb{C} \setminus \{0, -1, \ldots\}\right)$
$s \in \mathbb{Z}_{\ge 2} \;\implies\; \left(\zeta\!\left(s, a\right) \text{ is meromorphic on } a \in \mathbb{C}\right)$
$s \in \{0, -1, \ldots\} \;\implies\; \left(\zeta\!\left(s, a\right) \text{ is holomorphic on } a \in \mathbb{C}\right)$

## Specific values

$\zeta\!\left(s, 1\right) = \zeta\!\left(s\right)$
$\zeta\!\left(s, 2\right) = \zeta\!\left(s\right) - 1$
$\zeta\!\left(s, 3\right) = \zeta\!\left(s\right) - 1 - \frac{1}{{2}^{s}}$
$\zeta\!\left(s, n\right) = \zeta\!\left(s\right) - \sum_{k=1}^{n - 1} \frac{1}{{k}^{s}}$
$\zeta\!\left(s, \frac{1}{2}\right) = \left({2}^{s} - 1\right) \zeta\!\left(s\right)$
$\zeta\!\left(s, \frac{3}{2}\right) = \left({2}^{s} - 1\right) \zeta\!\left(s\right) - {2}^{s}$
$\zeta\!\left(s, \frac{1}{2} + n\right) = \left({2}^{s} - 1\right) \zeta\!\left(s\right) - {2}^{s} \sum_{k=0}^{n - 1} \frac{1}{{\left(2 k + 1\right)}^{s}}$
$\zeta\!\left(s, \frac{1}{4}\right) + \zeta\!\left(s, \frac{3}{4}\right) = {2}^{s} \left({2}^{s} - 1\right) \zeta\!\left(s\right)$
$\zeta\!\left(s, \frac{1}{6}\right) + \zeta\!\left(s, \frac{5}{6}\right) = \left({2}^{s} - 1\right) \left({3}^{s} - 1\right) \zeta\!\left(s\right)$
$\zeta\!\left(2, 1\right) = \frac{{\pi}^{2}}{6}$
$\zeta\!\left(2, 2\right) = \frac{{\pi}^{2}}{6} - 1$
$\zeta\!\left(3, 1\right) = \zeta\!\left(3\right)$
$\zeta\!\left(3, 2\right) = \zeta\!\left(3\right) - 1$
$\zeta\!\left(4, 1\right) = \frac{{\pi}^{4}}{90}$
$\zeta\!\left(4, 2\right) = \frac{{\pi}^{4}}{90} - 1$
$\zeta\!\left(2, \frac{1}{2}\right) = \frac{{\pi}^{2}}{2}$
$\zeta\!\left(3, \frac{1}{2}\right) = 7 \zeta\!\left(3\right)$
$\zeta\!\left(4, \frac{1}{2}\right) = \frac{{\pi}^{4}}{6}$
$\zeta\!\left(2, \frac{1}{4}\right) = {\pi}^{2} + 8 G$
$\zeta\!\left(2, \frac{3}{4}\right) = {\pi}^{2} - 8 G$
$\zeta\!\left(3, \frac{1}{4}\right) = 28 \zeta\!\left(3\right) + {\pi}^{3}$
$\zeta\!\left(3, \frac{3}{4}\right) = 28 \zeta\!\left(3\right) - {\pi}^{3}$
$\zeta\!\left(3, \frac{1}{6}\right) = 91 \zeta\!\left(3\right) + 2 \sqrt{3} {\pi}^{3}$
$\zeta\!\left(3, \frac{5}{6}\right) = 91 \zeta\!\left(3\right) - 2 \sqrt{3} {\pi}^{3}$
$\zeta\!\left(n, a\right) = \frac{{\left(-1\right)}^{n}}{\left(n - 1\right)!} \psi^{(n - 1)}\!\left(a\right)$
$\zeta\!\left(1, a\right) = {\tilde \infty}$
$\zeta\!\left(-n, a\right) = -\frac{B_{n + 1}\!\left(a\right)}{n + 1}$
$\zeta\!\left(-n, 0\right) = -\frac{B_{n + 1}}{n + 1}$
$\zeta\!\left(0, a\right) = \frac{1}{2} - a$
$\zeta\!\left(0, 0\right) = \frac{1}{2}$
$\zeta\!\left(0, \frac{1}{2}\right) = 0$
$\zeta\!\left(2, a\right) = {a}^{-2} \,{}_3F_2\!\left(1, a, a, a + 1, a + 1, 1\right)$

## Series representations

### Dirichlet series

$\zeta\!\left(s, a\right) = \sum_{n=0}^{\infty} \frac{1}{{\left(n + a\right)}^{s}}$
$\zeta^{(r)}\!\left(s, a\right) = {\left(-1\right)}^{r} \sum_{n=0}^{\infty} \frac{\log^{r}\!\left(n + a\right)}{{\left(n + a\right)}^{s}}$
$\zeta\!\left(s, N\right) = \sum_{n=N}^{\infty} \frac{1}{{n}^{s}}$

### Laurent series

Related topic: Stieltjes constants
$\zeta\!\left(s, a\right) = \frac{1}{s - 1} + \sum_{n=0}^{\infty} \frac{{\left(-1\right)}^{n}}{n !} \gamma_{n}\!\left(a\right) {\left(s - 1\right)}^{n}$

## Integral representations

$\zeta\!\left(s, a\right) = \frac{\pi}{2 \left(s - 1\right)} \int_{-\infty}^{\infty} \frac{{\left(a - \frac{1}{2} + i x\right)}^{1 - s}}{\cosh^{2}\!\left(\pi x\right)} \, dx$
$\zeta\!\left(s, a\right) = \frac{1}{\Gamma(s)} \int_{0}^{\infty} \frac{{x}^{s - 1} {e}^{-a x}}{1 - {e}^{-x}} \, dx$

## Functional equations

### Recurrence relations

$\zeta\!\left(s, a + 1\right) = \zeta\!\left(s, a\right) - \frac{1}{{a}^{s}}$
$\zeta\!\left(s, a + N\right) = \zeta\!\left(s, a\right) - \sum_{n=0}^{N - 1} \frac{1}{{\left(n + a\right)}^{s}}$
$\zeta^{(r)}\!\left(s, a + N\right) = \zeta^{(r)}\!\left(s, a\right) + {\left(-1\right)}^{r + 1} \sum_{k=0}^{N - 1} \frac{\log^{r}\!\left(a + k\right)}{{\left(a + k\right)}^{s}}$

### Multiplication formula

$\zeta\!\left(s, N a\right) = \frac{1}{{N}^{s}} \sum_{k=0}^{N - 1} \zeta\!\left(s, a + \frac{k}{N}\right)$
$\zeta\!\left(s, a\right) = \frac{1}{{2}^{s}} \left(\zeta\!\left(s, \frac{a}{2}\right) + \zeta\!\left(s, \frac{a + 1}{2}\right)\right)$
$\zeta\!\left(s, a\right) = \frac{1}{{N}^{s}} \sum_{k=0}^{N - 1} \zeta\!\left(s, \frac{a + k}{N}\right)$

### Reflection formula

$\zeta\!\left(1 - s, \frac{p}{q}\right) = \frac{2 \Gamma(s)}{{\left(2 \pi q\right)}^{s}} \sum_{k=1}^{q} \cos\!\left(\frac{\pi s}{2} - \frac{2 \pi k p}{q}\right) \zeta\!\left(s, \frac{k}{q}\right)$

## Derivatives and differential equations

### Argument derivatives

$\frac{d}{d s}\, \zeta\!\left(s, a\right) = \zeta'\!\left(s, a\right)$
$\frac{d^{r}}{{d s}^{r}} \zeta\!\left(s, a\right) = \zeta^{(r)}\!\left(s, a\right)$

### Parameter derivatives

$\frac{d}{d a}\, \zeta\!\left(s, a\right) = -s \zeta\!\left(s + 1, a\right)$
$\frac{d^{r}}{{d a}^{r}} \zeta\!\left(s, a\right) = \left(1 - s - r\right)_{r} \zeta\!\left(s + r, a\right)$

## Euler-Maclaurin formula

$\zeta\!\left(s, a\right) = \sum_{k=0}^{N - 1} \frac{1}{{\left(a + k\right)}^{s}} + \frac{{\left(a + N\right)}^{1 - s}}{s - 1} + \frac{1}{{\left(a + N\right)}^{s}} \left(\frac{1}{2} + \sum_{k=1}^{M} \frac{B_{2 k}}{\left(2 k\right)!} \frac{\left(s\right)_{2 k - 1}}{{\left(a + N\right)}^{2 k - 1}}\right) - \int_{N}^{\infty} \frac{B_{2 M}\!\left(t - \left\lfloor t \right\rfloor\right)}{\left(2 M\right)!} \frac{\left(s\right)_{2 M}}{{\left(a + t\right)}^{s + 2 M}} \, dt$

## Representation of other functions

### Riemann zeta function

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

### Bernoulli polynomials

Related topic: Bernoulli numbers and polynomials
$B_{n}\!\left(z\right) = -n \zeta\!\left(1 - n, z\right)$

### Gamma and related functions

Related topics: Digamma function, Barnes G-function
$\Gamma(z) = \sqrt{2 \pi} \exp\!\left(\zeta'\!\left(0, z\right)\right)$
$\log \Gamma(z) = \zeta'\!\left(0, z\right) + \frac{\log\!\left(2 \pi\right)}{2}$
$\psi\!\left(z\right) = \lim_{s \to 1} \left[\frac{1}{s - 1} - \zeta\!\left(s, z\right)\right]$
$\psi^{(m)}\!\left(z\right) = {\left(-1\right)}^{m + 1} m ! \zeta\!\left(m + 1, z\right)$
$\log G(z) = \left(z - 1\right) \log \Gamma(z) - \zeta'\!\left(-1, z\right) + \zeta'(-1)$

### Dirichlet L-functions

Related topic: Dirichlet L-functions
$L\!\left(s, \chi\right) = \frac{1}{{q}^{s}} \sum_{k=1}^{q} \chi(k) \zeta\!\left(s, \frac{k}{q}\right)$
$\zeta\!\left(s, \frac{k}{q}\right) = \frac{{q}^{s}}{\varphi(q)} \sum_{\chi \in G_{q}} \overline{\chi(k)} L\!\left(s, \chi\right)$

### Polylogarithms

$\operatorname{Li}_{s}\!\left(z\right) = \frac{\Gamma\!\left(1 - s\right)}{{\left(2 \pi\right)}^{1 - s}} \left({i}^{1 - s} \zeta\!\left(1 - s, \frac{1}{2} + \frac{\log\!\left(-z\right)}{2 \pi i}\right) + {i}^{s - 1} \zeta\!\left(1 - s, \frac{1}{2} - \frac{\log\!\left(-z\right)}{2 \pi i}\right)\right)$

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

2021-03-15 19:12:00.328586 UTC