gives the Hurwitz zeta function TemplateBox[{s, a}, HurwitzZeta].


  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • The Hurwitz zeta function is defined as an analytic continuation of TemplateBox[{s, a}, HurwitzZeta]=sum_(k=0)^(infty)(k+a)^(-s).
  • 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.


open allclose all

Basic Examples  (6)

Evaluate numerically:

Plot over a subset of the reals:

Plot over a subset of the complexes:

Series expansion at the origin:

Series expansion at Infinity:

Series expansion at a singular point:

Scope  (33)

Numerical Evaluation  (4)

Evaluate numerically:

Evaluate to high precision:

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

Complex number input:

Evaluate efficiently at high precision:

Specific Values  (5)

Simple exact values are generated automatically:

HurwitzZeta[s,a] for symbolic a:

HurwitzZeta[s,a] for symbolic s:

Values at zero:

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

Visualization  (3)

Plot the HurwitzZeta as a function of its parameter s:

Plot the HurwitzZeta function for various orders:

Plot the real part of HurwitzZeta function:

Plot the imaginary part of HurwitzZeta function:

Function Properties  (11)

Real domain of TemplateBox[{x, a}, HurwitzZeta]:

For positive , this is simply :

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

Complex domain:

For positive , this is again :

Approximate function range of TemplateBox[{x, 3}, HurwitzZeta]:

HurwitzZeta threads elementwise over lists:

HurwitzZeta is not an analytic function:

Nor is it meromorphic:

TemplateBox[{x, 3}, HurwitzZeta] is neither non-decreasing nor non-increasing:

TemplateBox[{x, 3}, HurwitzZeta] is not injective:

TemplateBox[{3, a}, HurwitzZeta] is surjective:

TemplateBox[{4, a}, HurwitzZeta] is not surjective:

TemplateBox[{x, 3}, HurwitzZeta] is neither non-negative nor non-positive:

TemplateBox[{2, a}, HurwitzZeta] has both singularity and discontinuity for negative integers:

TemplateBox[{x, 3}, HurwitzZeta] is neither convex nor concave:

TraditionalForm formatting:

Differentiation  (3)

First derivative with respect to a:

Higher derivatives with respect to a:

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

Formula for the ^(th) derivative with respect to a:

Integration  (3)

Compute the indefinite integral using Integrate:

Verify the anti-derivative:

Definite integral:

More integrals:

Series Expansions  (2)

Find the Taylor expansion using Series:

Plots of the first three approximations around :

Taylor expansion at a generic point:

Function Identities and Simplifications  (2)

HurwitzZeta is defined through the identity:

Recurrence identity:

Applications  (1)

The word count in a text follows a Zipf distribution:

Fit a ZipfDistribution to the word frequency data:

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

Visualize the CDFs up to the truncation value:

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

Properties & Relations  (2)

HurwitzZeta can be generated by symbolic solvers:

For , two-argument Zeta coincides with HurwitzZeta:

Possible Issues  (2)

HurwitzZeta differs from the two-argument form of Zeta by a different choice of branch cut:

HurwitzZeta includes singular terms, unlike Zeta:

Neat Examples  (1)

ComplexPlot of HurwitzZeta function, as a function of with :

Wolfram Research (2008), HurwitzZeta, Wolfram Language function,


Wolfram Research (2008), HurwitzZeta, Wolfram Language function,


Wolfram Language. 2008. "HurwitzZeta." Wolfram Language & System Documentation Center. Wolfram Research.


Wolfram Language. (2008). HurwitzZeta. Wolfram Language & System Documentation Center. Retrieved from


@misc{reference.wolfram_2024_hurwitzzeta, author="Wolfram Research", title="{HurwitzZeta}", year="2008", howpublished="\url{}", note=[Accessed: 26-May-2024 ]}


@online{reference.wolfram_2024_hurwitzzeta, organization={Wolfram Research}, title={HurwitzZeta}, year={2008}, url={}, note=[Accessed: 26-May-2024 ]}