HurwitzZeta
✖
HurwitzZeta
Details

- Mathematical function, suitable for both symbolic and numerical manipulation.
- The Hurwitz zeta function is defined as an analytic continuation of
.
- HurwitzZeta is identical to Zeta for
.
- Unlike Zeta, HurwitzZeta has singularities at
for non-negative integers
.
- HurwitzZeta has branch cut discontinuities in the complex
plane running from
to
.
- For certain special arguments, HurwitzZeta automatically evaluates to exact values.
- HurwitzZeta can be evaluated to arbitrary numerical precision.
- HurwitzZeta automatically threads over lists.
Examples
open allclose allBasic Examples (6)Summary of the most common use cases

https://wolfram.com/xid/0cptz1yjxn2waa-hqx2cr

Plot over a subset of the reals:

https://wolfram.com/xid/0cptz1yjxn2waa-xeo16

Plot over a subset of the complexes:

https://wolfram.com/xid/0cptz1yjxn2waa-kiedlx

Series expansion at the origin:

https://wolfram.com/xid/0cptz1yjxn2waa-fdkkja

Series expansion at Infinity:

https://wolfram.com/xid/0cptz1yjxn2waa-20imb

Series expansion at a singular point:

https://wolfram.com/xid/0cptz1yjxn2waa-d2klx1

Scope (35)Survey of the scope of standard use cases
Numerical Evaluation (6)

https://wolfram.com/xid/0cptz1yjxn2waa-l274ju


https://wolfram.com/xid/0cptz1yjxn2waa-cksbl4


https://wolfram.com/xid/0cptz1yjxn2waa-b0wt9

The precision of the output tracks the precision of the input:

https://wolfram.com/xid/0cptz1yjxn2waa-y7k4a


https://wolfram.com/xid/0cptz1yjxn2waa-k6bv9

Evaluate efficiently at high precision:

https://wolfram.com/xid/0cptz1yjxn2waa-di5gcr


https://wolfram.com/xid/0cptz1yjxn2waa-bq2c6r

Compute average-case statistical intervals using Around:

https://wolfram.com/xid/0cptz1yjxn2waa-cw18bq

Compute the elementwise values of an array:

https://wolfram.com/xid/0cptz1yjxn2waa-thgd2

Or compute the matrix HurwitzZeta function using MatrixFunction:

https://wolfram.com/xid/0cptz1yjxn2waa-o5jpo

Specific Values (5)
Simple exact values are generated automatically:

https://wolfram.com/xid/0cptz1yjxn2waa-nww7l

HurwitzZeta[s,a] for symbolic a:

https://wolfram.com/xid/0cptz1yjxn2waa-fc9m8o


https://wolfram.com/xid/0cptz1yjxn2waa-eew9ix

HurwitzZeta[s,a] for symbolic s:

https://wolfram.com/xid/0cptz1yjxn2waa-h7p5ce


https://wolfram.com/xid/0cptz1yjxn2waa-bmqd0y

Find a value of s for which HurwitzZeta[s,1]=1.05:

https://wolfram.com/xid/0cptz1yjxn2waa-f2hrld


https://wolfram.com/xid/0cptz1yjxn2waa-g33sup

Visualization (3)
Plot the HurwitzZeta as a function of its parameter s:

https://wolfram.com/xid/0cptz1yjxn2waa-b1j98m

Plot the HurwitzZeta function for various orders:

https://wolfram.com/xid/0cptz1yjxn2waa-ecj8m7

Plot the real part of HurwitzZeta function:

https://wolfram.com/xid/0cptz1yjxn2waa-b8vnf

Plot the imaginary part of HurwitzZeta function:

https://wolfram.com/xid/0cptz1yjxn2waa-g3i7d1

Function Properties (11)

https://wolfram.com/xid/0cptz1yjxn2waa-z9olij

For positive , this is simply
:

https://wolfram.com/xid/0cptz1yjxn2waa-kcpp56

For negative integer , the domain is just the negative integers:

https://wolfram.com/xid/0cptz1yjxn2waa-e6vdr5


https://wolfram.com/xid/0cptz1yjxn2waa-c1mghp

For positive , this is again
:

https://wolfram.com/xid/0cptz1yjxn2waa-mdpxud

Approximate function range of :

https://wolfram.com/xid/0cptz1yjxn2waa-rpof25

HurwitzZeta threads elementwise over lists:

https://wolfram.com/xid/0cptz1yjxn2waa-dtmf5x

HurwitzZeta is not an analytic function:

https://wolfram.com/xid/0cptz1yjxn2waa-h5x4l2


https://wolfram.com/xid/0cptz1yjxn2waa-89mg0d

is neither non-decreasing nor non-increasing:

https://wolfram.com/xid/0cptz1yjxn2waa-kkey2o


https://wolfram.com/xid/0cptz1yjxn2waa-jw54pe


https://wolfram.com/xid/0cptz1yjxn2waa-ctca0g


https://wolfram.com/xid/0cptz1yjxn2waa-hkqec4


https://wolfram.com/xid/0cptz1yjxn2waa-1fwuk8


https://wolfram.com/xid/0cptz1yjxn2waa-hdm869

is neither non-negative nor non-positive:

https://wolfram.com/xid/0cptz1yjxn2waa-84dui

has both singularity and discontinuity for negative integers:

https://wolfram.com/xid/0cptz1yjxn2waa-mdtl3h


https://wolfram.com/xid/0cptz1yjxn2waa-mn5jws

is neither convex nor concave:

https://wolfram.com/xid/0cptz1yjxn2waa-2xfltv

TraditionalForm formatting:

https://wolfram.com/xid/0cptz1yjxn2waa-daq9d3

Differentiation (3)
First derivative with respect to a:

https://wolfram.com/xid/0cptz1yjxn2waa-krpoah

Higher derivatives with respect to a:

https://wolfram.com/xid/0cptz1yjxn2waa-z33jv

Plot the higher derivatives with respect to a when s=3:

https://wolfram.com/xid/0cptz1yjxn2waa-fxwmfc

Formula for the derivative with respect to a:

https://wolfram.com/xid/0cptz1yjxn2waa-cb5zgj

Integration (3)
Compute the indefinite integral using Integrate:

https://wolfram.com/xid/0cptz1yjxn2waa-bponid


https://wolfram.com/xid/0cptz1yjxn2waa-op9yly


https://wolfram.com/xid/0cptz1yjxn2waa-b9jw7l


https://wolfram.com/xid/0cptz1yjxn2waa-4nbst


https://wolfram.com/xid/0cptz1yjxn2waa-cas

Series Expansions (2)
Find the Taylor expansion using Series:

https://wolfram.com/xid/0cptz1yjxn2waa-ewr1h8

Plots of the first three approximations around :

https://wolfram.com/xid/0cptz1yjxn2waa-binhar

Taylor expansion at a generic point:

https://wolfram.com/xid/0cptz1yjxn2waa-jwxla7

Function Identities and Simplifications (2)
HurwitzZeta is defined through the identity:

https://wolfram.com/xid/0cptz1yjxn2waa-b5js3b


https://wolfram.com/xid/0cptz1yjxn2waa-e1c09w

Applications (1)Sample problems that can be solved with this function
The word count in a text follows a Zipf distribution:

https://wolfram.com/xid/0cptz1yjxn2waa-ir5i5j

https://wolfram.com/xid/0cptz1yjxn2waa-z0k8fd
Fit a ZipfDistribution to the word frequency data:

https://wolfram.com/xid/0cptz1yjxn2waa-798qy3

Fit a truncated ZipfDistribution to counts at most 50 using rhohat as a starting value:

https://wolfram.com/xid/0cptz1yjxn2waa-hxqqsw

Visualize the CDFs up to the truncation value:

https://wolfram.com/xid/0cptz1yjxn2waa-enj6pg

Estimate the proportion of the original data not included in the truncated model:

https://wolfram.com/xid/0cptz1yjxn2waa-cxgs8z


https://wolfram.com/xid/0cptz1yjxn2waa-lndm5

Properties & Relations (2)Properties of the function, and connections to other functions
HurwitzZeta can be generated by symbolic solvers:

https://wolfram.com/xid/0cptz1yjxn2waa-dw0258


https://wolfram.com/xid/0cptz1yjxn2waa-ijuegg

For , two-argument Zeta coincides with HurwitzZeta:

https://wolfram.com/xid/0cptz1yjxn2waa-bg1kx5

Possible Issues (2)Common pitfalls and unexpected behavior
HurwitzZeta differs from the two-argument form of Zeta by a different choice of branch cut:

https://wolfram.com/xid/0cptz1yjxn2waa-e9pv0p


https://wolfram.com/xid/0cptz1yjxn2waa-bffrvn


https://wolfram.com/xid/0cptz1yjxn2waa-gcw9zv


https://wolfram.com/xid/0cptz1yjxn2waa-ewvvvw

HurwitzZeta includes singular terms, unlike Zeta:

https://wolfram.com/xid/0cptz1yjxn2waa-i9iko1


https://wolfram.com/xid/0cptz1yjxn2waa-cpzivg


https://wolfram.com/xid/0cptz1yjxn2waa-biasap

Neat Examples (1)Surprising or curious use cases
ComplexPlot of HurwitzZeta function, as a function of with
:

https://wolfram.com/xid/0cptz1yjxn2waa-i6fcaw

Wolfram Research (2008), HurwitzZeta, Wolfram Language function, https://reference.wolfram.com/language/ref/HurwitzZeta.html.
Text
Wolfram Research (2008), HurwitzZeta, Wolfram Language function, https://reference.wolfram.com/language/ref/HurwitzZeta.html.
Wolfram Research (2008), HurwitzZeta, Wolfram Language function, https://reference.wolfram.com/language/ref/HurwitzZeta.html.
CMS
Wolfram Language. 2008. "HurwitzZeta." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/HurwitzZeta.html.
Wolfram Language. 2008. "HurwitzZeta." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/HurwitzZeta.html.
APA
Wolfram Language. (2008). HurwitzZeta. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/HurwitzZeta.html
Wolfram Language. (2008). HurwitzZeta. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/HurwitzZeta.html
BibTeX
@misc{reference.wolfram_2025_hurwitzzeta, author="Wolfram Research", title="{HurwitzZeta}", year="2008", howpublished="\url{https://reference.wolfram.com/language/ref/HurwitzZeta.html}", note=[Accessed: 08-June-2025
]}
BibLaTeX
@online{reference.wolfram_2025_hurwitzzeta, organization={Wolfram Research}, title={HurwitzZeta}, year={2008}, url={https://reference.wolfram.com/language/ref/HurwitzZeta.html}, note=[Accessed: 08-June-2025
]}