# Hurwitz Zeta Function

## Definition

$\displaystyle {\displaystyle \HurwitzZeta@{s}{a} = \sum_{n=0}^\infty \frac{1}{(n+a)^s} }$

Constraint(s): $\displaystyle {\displaystyle \realpart{s} > 1}$ &
$\displaystyle {\displaystyle a \neq 0,-1,-2,\dots}$

$\displaystyle {\displaystyle \HurwitzZeta@{s}{1} = \RiemannZeta@{s} }$
$\displaystyle {\displaystyle \HurwitzZeta@{s}{a} = \HurwitzZeta@{s}{a+1} + a^{-s} }$
$\displaystyle {\displaystyle \HurwitzZeta@{s}{a} = \HurwitzZeta@{s}{a+m} + \sum_{n=0}^{m-1} \frac{1}{(n+a)^s} }$

Constraint(s): $\displaystyle {\displaystyle m = 1,2,3,\dots}$

## Representations by the Euler-Maclaurin Formula

$\displaystyle {\displaystyle \HurwitzZeta@{s}{a} = \sum_{n=0}^N \frac{1}{(n+a)^s} + \frac{(N+a)^{1-s}}{s-1} - s \int_N^\infty \frac{x-\floor{x}}{(x+a)^{s+1}} \diff{x} }$

Constraint(s): $\displaystyle {\displaystyle s \neq 1}$ &
$\displaystyle {\displaystyle \realpart{s} > 0}$ &
$\displaystyle {\displaystyle a > 0}$ &
$\displaystyle {\displaystyle N = 0,1,2,3,\dots}$

$\displaystyle {\displaystyle \HurwitzZeta@{s}{a} = \frac{1}{a^s} \left( \frac{1}{2} + \frac{a}{s-1} \right) - s (s+1) \int_0^\infty \frac{\PeriodicBernoulliB{2}@{x}}{(x+a)^{s+2}} \diff{x} }$

Constraint(s): $\displaystyle {\displaystyle s \neq 1}$ &
$\displaystyle {\displaystyle \realpart{s} > -1}$ &
$\displaystyle {\displaystyle a > 0}$

$\displaystyle {\displaystyle \HurwitzZeta@{s}{a} = \frac{1}{a^s} + \frac{1}{(1+a)^s} \left( \frac{1}{2} + \frac{1+a}{s-1} \right) + \sum_{k=1}^n \binom{s+2k-2}{2k-1} \frac{\BernoulliB{2k}}{2k} \frac{1}{(1+a)^{s+2k-1}} - \binom{s+2n}{2n+1} \int_1^\infty \frac{\PeriodicBernoulliB{2n+1}@{x}}{(x+a)^{s+2n+1}} \diff{x} }$

Constraint(s): $\displaystyle {\displaystyle s \neq 1}$ &
$\displaystyle {\displaystyle a > 0}$ &
$\displaystyle {\displaystyle n = 1,2,3,\dots}$ &
$\displaystyle {\displaystyle \realpart{s} > -2n}$

## Series Representations

$\displaystyle {\displaystyle \HurwitzZeta@{s}{\tfrac{1}{2} a} = \HurwitzZeta@{s}{\tfrac{1}{2} a + \tfrac{1}{2}} + 2^s \sum_{n=0}^\infty \frac{\opminus^n}{(n+a)^s} }$

Constraint(s): $\displaystyle {\displaystyle \realpart{s} > 0}$ &
$\displaystyle {\displaystyle s \neq 1}$ &
$\displaystyle {\displaystyle 0 < a \leq 1}$

$\displaystyle {\displaystyle \HurwitzZeta@{1-s}{a} = \frac{2\EulerGamma@{s}}{(2\cpi)^s} \* \sum_{n=1}^\infty \frac{1}{n^s} \cos@{\tfrac{1}{2} \cpi s - 2n \cpi a} }$

Constraint(s): $\displaystyle {\displaystyle \realpart{s} > 1}$ &
$\displaystyle {\displaystyle 0 < a \leq 1}$

$\displaystyle {\displaystyle \HurwitzZeta@{s}{a} = \sum_{n=0}^\infty \frac{\EulerGamma@{n+s}}{n! \EulerGamma@{s}} \RiemannZeta@{n+s} (1-a)^n }$

Constraint(s): $\displaystyle {\displaystyle s \neq 1}$ &
$\displaystyle {\displaystyle |a-1| < 1}$

## Special Values

$\displaystyle {\displaystyle \HurwitzZeta@{s}{\tfrac{1}{2}} = (2^s - 1) \RiemannZeta@{s} }$

Constraint(s): $\displaystyle {\displaystyle s \neq 1}$

$\displaystyle {\displaystyle \HurwitzZeta@{n+1}{a} = \frac{\opminus^{n+1} \digamma^{(n)}@{a}}{n!} }$

Constraint(s): $\displaystyle {\displaystyle n = 1,2,3,\dots}$ &
$\displaystyle {\displaystyle \realpart{a} > 0}$

$\displaystyle {\displaystyle \HurwitzZeta@{0}{a} = \tfrac{1}{2} - a }$

Constraint(s): $\displaystyle {\displaystyle \realpart{a} > 0}$

$\displaystyle {\displaystyle \HurwitzZeta@{-n}{a} = -\frac{\BernoulliB{n+1}@{a}}{n+1} }$

Constraint(s): $\displaystyle {\displaystyle n = 0,1,2,\dots}$ &
$\displaystyle {\displaystyle \realpart{a} > 0}$

$\displaystyle {\displaystyle \HurwitzZeta@{s}{ka} = k^{-s} \* \sum_{n=0}^{k-1} \HurwitzZeta@{s}{a+\frac{n}{k}} }$

Constraint(s): $\displaystyle {\displaystyle s \neq 1}$ &
$\displaystyle {\displaystyle k = 1,2,3,\dots}$ &
$\displaystyle {\displaystyle \realpart{a} > 0}$

$\displaystyle {\displaystyle \HurwitzZeta@{1-s}{\frac{h}{k}} = \frac{2 \EulerGamma@{s}}{(2 \cpi k)^s} \* \sum_{r=1}^k \cos@{\frac{\cpi s}{2} - \frac{2 \cpi r h}{k}} \HurwitzZeta@{s}{\frac{r}{k}} }$

Constraint(s): $\displaystyle {\displaystyle s \neq 0,1}$ &
$\displaystyle {\displaystyle h,k}$ integers, $\displaystyle {\displaystyle 1 \leq h \leq k}$ &
$\displaystyle {\displaystyle \realpart{a} > 0}$

## Derivatives

### a-Derivative

$\displaystyle {\displaystyle \pderiv{}{a} \HurwitzZeta@{s}{a} = -s \HurwitzZeta@{s+1}{a} }$

Constraint(s): $\displaystyle {\displaystyle s \neq 0,1}$ &
$\displaystyle {\displaystyle \realpart{a} > 0}$

### s-Derivatives

$\displaystyle {\displaystyle \HurwitzZeta'@{0}{a} = \ln@@{\EulerGamma@{a}} - \tfrac{1}{2} \ln@{2\cpi} }$

Constraint(s): $\displaystyle {\displaystyle a > 0}$

$\displaystyle {\displaystyle \HurwitzZeta'@{s}{a} = -\frac{\ln@@{a}}{a^s} \left( \frac{1}{2} + \frac{a}{s-1} \right) - \frac{a^{1-s}}{(s-1)^2} + s (s+1) \int_0^\infty \frac{\PeriodicBernoulliB{2}@{x} \ln@{x+a}}{(x+a)^{s+2}} \diff{x} - (2s+1) \int_0^\infty \frac{\PeriodicBernoulliB{2}@{x}}{(x+a)^{s+2}} \diff{x} }$

Constraint(s): $\displaystyle {\displaystyle \realpart{s} > -1}$ &
$\displaystyle {\displaystyle s \neq 1}$ &
$\displaystyle {\displaystyle a > 0}$

$\displaystyle {\displaystyle \opminus^k \HurwitzZeta^{(k)}@{s}{a} = \frac{(\ln@@{a})^k}{a^s} \left( \frac{1}{2} + \frac{a}{s-1} \right) + k! a^{1-s} \sum_{r=0}^{k-1} \frac{(\ln@@{a})^r}{r! (s-1)^{k-r+1}} - s (s+1) \int_0^\infty \frac{\PeriodicBernoulliB{2}@{x} (\ln@{x+a})^k} {(x+a)^{s+2}} \diff{x} + k (2s+1) \int_0^\infty \frac{\PeriodicBernoulliB{2}@{x} (\ln@{x+a})^{k-1}} {(x+a)^{s+2}} \diff{x} - k (k-1) \int_0^\infty \frac{\PeriodicBernoulliB{2}@{x} (\ln@{x+a})^{k-2}} {(x+a)^{s+2}} \diff{x} }$

Constraint(s): $\displaystyle {\displaystyle \realpart{s} > -1}$ &
$\displaystyle {\displaystyle s \neq 1}$ &
$\displaystyle {\displaystyle a > 0}$

$\displaystyle {\displaystyle \HurwitzZeta'@{1-2n}{\frac{h}{k}} = \frac{(\digamma@{2n} - \ln@{2 \cpi k}) \BernoulliB{2n}@{h/k}}{2n} - \frac{(\digamma@{2n} - \ln@{2 \cpi}) \BernoulliB{2n}}{2 n k^{2n}} + \frac{\opminus^{n+1} \cpi}{(2 \cpi k)^{2n}} \sum_{r=1}^{k-1} \sin@{\frac{2 \cpi r h}{k}} \digamma^{(2n-1)}@{\frac{r}{k}} + \frac{\opminus^{n+1} 2 \cdot (2n-1)!}{(2 \cpi k)^{2n}} \sum_{r=1}^{k-1} \cos@{\frac{2 \cpi r h}{k}} \HurwitzZeta'@{2n}{\frac{r}{k}} + \frac{\RiemannZeta'@{1-2n}}{k^{2n}} }$

Constraint(s): $\displaystyle {\displaystyle h,k}$ are integers with $\displaystyle {\displaystyle 1 \leq h \leq k}$ &
$\displaystyle {\displaystyle n = 1,2,3,\dots}$

$\displaystyle {\displaystyle \HurwitzZeta'@{1-2n}{\tfrac{1}{2}} = - \frac{\BernoulliB{2n} \ln@@{2}}{n \cdot 4^n} - \frac{(2^{2n-1}-1) \RiemannZeta'@{1-2n}}{2^{2n-1}} }$

Constraint(s): $\displaystyle {\displaystyle n = 1,2,3,\dots}$

$\displaystyle {\displaystyle \HurwitzZeta'@{1-2n}{\tfrac{1}{3}} = -\frac{\cpi (9^n-1) \BernoulliB{2n}}{8n \sqrt{3} (3^{2n-1}-1)} - \frac{\BernoulliB{2n} \ln@@{3}}{4n \cdot 3^{2n-1}} - \frac{\opminus^n \digamma^{(2n-1)}@{\frac{1}{3}}}{2 \sqrt{3} (6\cpi)^{2n-1}} - \frac{\left( 3^{2n-1}-1 \right) \RiemannZeta'@{1-2n}}{2 \cdot 3^{2n-1}} }$

Constraint(s): $\displaystyle {\displaystyle n = 1,2,3,\dots}$

$\displaystyle {\displaystyle \sum_{r \hiderel{=} 1}^{k-1} \HurwitzZeta'@{s}{\frac{r}{k}} = (k^s - 1) \RiemannZeta'@{s} + k^s \RiemannZeta@{s} \ln@@{k} }$

Constraint(s): $\displaystyle {\displaystyle s \neq 1}$ &
$\displaystyle {\displaystyle k = 1,2,3,\dots}$

## Integral Representations

$\displaystyle {\displaystyle \HurwitzZeta@{s}{a} = \frac{1}{\EulerGamma@{s}} \int_0^\infty \frac{x^{s-1} \expe^{-ax}}{1-\expe^{-x}} \diff{x} }$

Constraint(s): $\displaystyle {\displaystyle \realpart{s} > 1}$ &
$\displaystyle {\displaystyle \realpart{a} > 0}$

$\displaystyle {\displaystyle \HurwitzZeta@{s}{a} = -s \int_{-a}^\infty \frac{x-\floor{x}-\frac{1}{2}}{(x+a)^{s+1}} \diff{x} }$

Constraint(s): $\displaystyle {\displaystyle -1 < \realpart{s} < 0}$ &
$\displaystyle {\displaystyle 0 < a \leq 1}$

$\displaystyle {\displaystyle \HurwitzZeta@{s}{a} = \frac{1}{2} a^{-s} + \frac{a^{1-s}}{s-1} + \frac{1}{\EulerGamma@{s}} \int_0^\infty \left( \frac{1}{\expe^x-1} - \frac{1}{x}+ \frac{1}{2} \right) \frac{x^{s-1}}{\expe^{ax}} \diff{x} }$

Constraint(s): $\displaystyle {\displaystyle \realpart{s} > -1}$ &
$\displaystyle {\displaystyle s \neq 1}$ &
$\displaystyle {\displaystyle \realpart{a} > 0}$

$\displaystyle {\displaystyle \HurwitzZeta@{s}{a} = \frac{1}{2} a^{-s}+ \frac{a^{1-s}}{s-1} + \sum_{k=1}^n \frac{\EulerGamma@{s+2k-1}}{\EulerGamma@{s}} \frac{\BernoulliB{2k}}{(2k)!} a^{-2k-s+1} + \frac{1}{\EulerGamma@{s}} \int_0^\infty \left( \frac{1}{\expe^x-1} - \frac{1}{x} + \frac{1}{2} - \sum_{k=1}^n \frac{\BernoulliB{2k}}{(2k)!} x^{2k-1} \right) x^{s-1} \expe^{-ax} \diff{x} }$

Constraint(s): $\displaystyle {\displaystyle \realpart{s} > -(2n+1)}$ &
$\displaystyle {\displaystyle s \neq 1}$ &
$\displaystyle {\displaystyle \realpart{a} > 0}$

$\displaystyle {\displaystyle \HurwitzZeta@{s}{a} = \frac{1}{2} a^{-s} + \frac{a^{1-s}}{s-1} + 2 \int_0^\infty \frac{\sin@{s \atan@{x/a}}}{(a^2+x^2)^{s/2} (\expe^{2 \cpi x}-1)} \diff{x} }$

Constraint(s): $\displaystyle {\displaystyle s \neq 1}$ &
$\displaystyle {\displaystyle \realpart{a} > 0}$

$\displaystyle {\displaystyle \HurwitzZeta@{s}{a} = \frac{\EulerGamma@{1-s}}{2 \cpi \iunit} \int_{-\infty}^{(0+)} \frac{\expe^{az} z^{s-1}}{1 - \expe^z} \diff{z} }$

Constraint(s): $\displaystyle {\displaystyle s \neq 1}$ &
$\displaystyle {\displaystyle \realpart{a} > 0}$ &

the integration contour is a loop around the negative real axis; it starts at $\displaystyle {\displaystyle -\infty}$ , encircles the origin once in the positive direction without enclosing any of the points

$\displaystyle {\displaystyle z=\pm2\cpi\iunit}$ , $\displaystyle {\displaystyle \pm4\cpi\iunit, \ldots,}$ and returns to $\displaystyle {\displaystyle -\infty}$

## Further Integral Representations

$\displaystyle {\displaystyle \frac{1}{\EulerGamma@{s}} \int_0^\infty \frac{x^{s-1} \expe^{-ax}}{2 \cosh@@{x}} \diff{x} = 4^{-s} \left( \HurwitzZeta@{s}{\tfrac{1}{4} + \tfrac{1}{4} a} - \HurwitzZeta@{s}{\tfrac{3}{4} + \tfrac{1}{4} a} \right) }$

Constraint(s): $\displaystyle {\displaystyle \realpart{s} > 0}$ &
$\displaystyle {\displaystyle \realpart{a} > -1}$

$\displaystyle {\displaystyle \int_0^a x^n \digamma@{x} \diff{x} = \opminus^{n-1} \RiemannZeta'@{-n} + \opminus^n \HarmonicNumber{n} \frac{\BernoulliB{n+1}}{n+1} - \sum_{k=0}^n \opminus^k \binom{n}{k} h(k) \frac{\BernoulliB{k+1}(a)}{k+1} a^{n-k} + \sum_{k=0}^n \opminus^k \binom{n}{k} \HurwitzZeta'@{-k}{a} a^{n-k} }$

Substitution(s): $\displaystyle {\displaystyle {\displaystyle \HarmonicNumber{n} = \sum_{k=1}^n k^{-1}}}$

Constraint(s): $\displaystyle {\displaystyle n = 1,2,\dots}$ &
$\displaystyle {\displaystyle \realpart{a} > 0}$

$\displaystyle {\displaystyle n \int_0^a \HurwitzZeta'@{1-n}{x} \diff{x} = \HurwitzZeta'@{-n}{a} - \RiemannZeta'@{-n} + \frac{\BernoulliB{n+1} - \BernoulliB{n+1}@{a}}{n(n+1)} }$

Constraint(s): $\displaystyle {\displaystyle n = 1,2,\dots}$ &
$\displaystyle {\displaystyle \realpart{a} > 0}$

## Further Series Representations

$\displaystyle {\displaystyle \sum_{n \hiderel{=} 0}^\infty \frac{\opminus^n}{(n+a)^s} \hiderel{=} \frac{1}{\EulerGamma@{s}} \int_0^\infty \frac{x^{s-1} \expe^{-ax}}{1+\expe^{-x}} \diff{x} \hiderel{=} 2^{-s} \left( \HurwitzZeta@{s}{\tfrac{1}{2}a} - \HurwitzZeta@{s}{\tfrac{1}{2} (1+a)} \right) }$

Constraint(s): $\displaystyle {\displaystyle \realpart{a} > 0}$ , $\displaystyle {\displaystyle \realpart{s} > 0}$ &
or $\displaystyle {\displaystyle \realpart{a} = 0}$ , $\displaystyle {\displaystyle \imagpart{a} \neq 0}$ , $\displaystyle {\displaystyle 0 < \realpart{s} < 1}$

$\displaystyle {\displaystyle \sum_{n \hiderel{=} 1}^\infty \frac{\chi(n)}{n^s} = k^{-s} \sum_{r=1}^k \chi(r) \HurwitzZeta@{s}{\frac{r}{k}} }$

Constraint(s): $\displaystyle {\displaystyle \realpart{s} > 1}$ &
$\displaystyle {\displaystyle \chi(n)}$ is a Dirichlet character~$\displaystyle {\displaystyle \pmod{k}}$

## Sums

$\displaystyle {\displaystyle \sum_{k \hiderel{=} 1}^\infty \frac{\opminus^k}{k} \HurwitzZeta@{nk}{a} = -n \ln@@{\EulerGamma@{a}} + \ln@{\prod_{j=0}^{n-1} \EulerGamma@{a-\expe^{(2j+1)\cpi \iunit/n}}} }$

Constraint(s): $\displaystyle {\displaystyle n = 2,3,4,\dots}$ &
$\displaystyle {\displaystyle \realpart{a} \geq 1}$

$\displaystyle {\displaystyle \sum_{k \hiderel{=} 1}^\infty \binom{n+k}{k} \HurwitzZeta@{n+k+1}{a} z^k = \frac{\opminus^n}{n!} \left( \digamma^{(n)}@{a} - \digamma^{(n)}@{a-z} \right) }$

Constraint(s): $\displaystyle {\displaystyle n = 1,2,3,\dots}$ &
$\displaystyle {\displaystyle \realpart{a} > 0}$ &
$\displaystyle {\displaystyle |z| < |a|}$

$\displaystyle {\displaystyle \sum_{k \hiderel{=} 2}^\infty \frac{k}{2^k} \HurwitzZeta@{k+1}{\tfrac{3}{4}} = 8\CatalansConstant }$

## a-Asymptotic Behavior

$\displaystyle {\displaystyle \HurwitzZeta@{s}{a+1} = \RiemannZeta@{s} - s \RiemannZeta@{s+1} a + \BigO@{a^2} }$

Constraint(s): $\displaystyle {\displaystyle a \to 0}$ with $\displaystyle {\displaystyle s}$ $\displaystyle {\displaystyle (\neq 1)}$ fixed

$\displaystyle {\displaystyle \HurwitzZeta@{s}{\alpha+\iunit\beta} \to 0 }$

Constraint(s): $\displaystyle {\displaystyle \beta \to \pm\infty}$ with $\displaystyle {\displaystyle s}$ fixed &
$\displaystyle {\displaystyle \realpart s > 1}$

$\displaystyle {\displaystyle \HurwitzZeta@{s}{a} - \frac{a^{1-s}}{s-1} - \frac{1}{2} a^{-s} \sim \sum_{k=1}^{\infty} \frac{\BernoulliB{2k}}{(2k)!} \frac{\EulerGamma@{s+2k-1}}{\EulerGamma@{s}} a^{1-s-2k} }$

Constraint(s): $\displaystyle {\displaystyle a \to \infty}$ in the sector $\displaystyle {\displaystyle |\ph@@{a}| \leq \cpi-\delta (<\cpi)}$ , with $\displaystyle {\displaystyle s (\neq 1)}$ and $\displaystyle {\displaystyle \delta}$ fixed

$\displaystyle {\displaystyle \HurwitzZeta'@{-1}{a} - \frac{1}{12} + \frac{1}{4} a^2 - \left( \frac{1}{12} - \frac{1}{2} a + \frac{1}{2} a^2 \right) \ln@@{a} \sim -\sum_{k=1}^\infty \frac{\BernoulliB{2k+2}}{(2k+2)(2k+1)2k} a^{-2k} }$

Constraint(s): $\displaystyle {\displaystyle a \to \infty}$ in the sector $\displaystyle {\displaystyle |\ph@@{a}| \leq \tfrac{1}{2} \cpi-\delta (<\tfrac{1}{2}\cpi)}$

$\displaystyle {\displaystyle \HurwitzZeta'@{-2}{a} - \frac{1}{12} a + \frac{1}{9} a^3 - \left( \frac{1}{6} a - \frac{1}{2} a^2 + \frac{1}{3} a^3 \right) \ln@@{a} \sim \sum_{k=1}^\infty \frac{2\!\BernoulliB{2k+2}}{(2k+2)(2k+1)2k(2k-1)} a^{-(2k-1)} }$