# Formula:DLMF:25.11:E7

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://vmext-demo.wmflabs.org/v1/":): {\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)

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://vmext-demo.wmflabs.org/v1/":): {\displaystyle {\displaystyle s \neq 1}}**&

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://vmext-demo.wmflabs.org/v1/":): {\displaystyle {\displaystyle a > 0}}**&

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://vmext-demo.wmflabs.org/v1/":): {\displaystyle {\displaystyle n = 1,2,3,\dots}}**&

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://vmext-demo.wmflabs.org/v1/":): {\displaystyle {\displaystyle \realpart{s} > -2n}}**

## Proof

We ask users to provide proof(s), reference(s) to proof(s), or further clarification on the proof(s) in this space.

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://vmext-demo.wmflabs.org/v1/":): {\displaystyle {\displaystyle N = 1}}**in

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://vmext-demo.wmflabs.org/v1/":): {\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} }}**

## Symbols List

& : logical and

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://vmext-demo.wmflabs.org/v1/":): {\displaystyle {\displaystyle \zeta}}**
: Hurwitz zeta function : http://dlmf.nist.gov/25.11#E1

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://vmext-demo.wmflabs.org/v1/":): {\displaystyle {\displaystyle \Sigma}}**
: sum : http://drmf.wmflabs.org/wiki/Definition:sum

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://vmext-demo.wmflabs.org/v1/":): {\displaystyle {\displaystyle \binom{n}{k}}}**
: binomial coefficient : http://dlmf.nist.gov/1.2#E1 http://dlmf.nist.gov/26.3#SS1.p1

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://vmext-demo.wmflabs.org/v1/":): {\displaystyle {\displaystyle B_{n}}}**
: Bernoulli polynomial : http://dlmf.nist.gov/24.2#i

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://vmext-demo.wmflabs.org/v1/":): {\displaystyle {\displaystyle \int}}**
: integral : http://dlmf.nist.gov/1.4#iv

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://vmext-demo.wmflabs.org/v1/":): {\displaystyle {\displaystyle \widetilde{B}_{n}}}**
: periodic Bernoulli functions : http://dlmf.nist.gov/24.2#iii

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://vmext-demo.wmflabs.org/v1/":): {\displaystyle {\displaystyle \mathrm{d}^nx}}**
: differential : http://dlmf.nist.gov/1.4#iv

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://vmext-demo.wmflabs.org/v1/":): {\displaystyle {\displaystyle \Re {z}}}**
: real part : http://dlmf.nist.gov/1.9#E2

## Bibliography

Equation (7), Section 25.11 of **DLMF**.

## URL links

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