Zeta
Details and Options
- Mathematical function, suitable for both symbolic and numerical manipulation.
- For Re(s)>1, TemplateBox[{s}, Zeta]=sum_(k=1)^inftyk^(-s).
- TemplateBox[{s, a}, Zeta2]=sum_(k=0)^(infty)(k+a)^(-s), where any term with is excluded.
- For Re(a)<0, the definition used is TemplateBox[{s, a}, Zeta2]=sum_(k=0)^(infty)((k+a)^2)^(-s/2).
- Zeta [s] has no branch cut discontinuities.
- For certain special arguments, Zeta automatically evaluates to exact values.
- Zeta can be evaluated to arbitrary numerical precision.
- Zeta automatically threads over lists.
- Zeta can be used with Interval and CenteredInterval objects. »
Examples
open all close allBasic Examples (6)
Evaluate numerically:
Generalized (Hurwitz) zeta function:
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 (36)
Numerical Evaluation (7)
Evaluate numerically:
Evaluate to high precision:
The precision of the output tracks the precision of the input:
Complex number inputs:
Evaluate efficiently at high precision:
Compute worst-case guaranteed intervals using Interval and CenteredInterval objects:
Compute average-case statistical intervals using Around :
Compute the elementwise values of an array:
Or compute the matrix Zeta function using MatrixFunction :
Specific Values (6)
Visualization (3)
Function Properties (12)
Real domain of Zeta :
Complex domain:
The generalized zeta function TemplateBox[{z, a}, Zeta2] has the same domain for all :
Zeta achieves all real values:
Zeta has the mirror property zeta (TemplateBox[{z}, Conjugate])=TemplateBox[{TemplateBox[{z}, Zeta]}, Conjugate]:
Zeta threads elementwise over lists and matrices:
Zeta is not an analytic function:
However, it is meromorphic:
Zeta is neither non-decreasing nor non-increasing:
However, it is decreasing to the right of the singularity at 1:
Zeta is not injective:
Zeta is surjective:
Zeta is neither non-negative nor non-positive:
TemplateBox[{x, a}, Zeta2] has both singularity and discontinuity at :
TemplateBox[{x}, Zeta] is neither convex nor concave:
However, it is convex to the right of the singularity at :
TraditionalForm formatting:
Differentiation (3)
First derivative with respect to :
Evaluate derivatives exactly for the Riemann zeta function:
Higher derivatives with respect to :
Plot the higher derivatives with respect to when :
Series Expansions (2)
Find the Taylor expansion using Series :
Plots of the first three approximations around :
General term in the series expansion using SeriesCoefficient :
Applications (7)
Plot the real part of the zeta function on the critical line:
Plot the real part across the critical strip:
Find a zero of the zeta function:
Find several zeros:
Use ZetaZero :
Find what fraction of pairs of the first 100 integers are relatively prime:
Compare with a zeta function formula:
Plot real and imaginary parts in the vicinity of two very nearby zeros (a Lehmer pair):
Plot the generalized zeta function:
Use MellinTransform to find the first two terms in the asymptotic expansion for a function that is defined by an infinite series:
Compute the Mellin transform of :
Compute the residues at and to obtain the required asymptotic expansion represented with Zeta function:
Properties & Relations (8)
Riemann Zeta Function (5)
The defining sum for the zeta function:
The Euler product formula for the zeta function:
Sum involving a zeta function:
Use FullSimplify to prove the functional equation:
Zeta can be represented as a DifferenceRoot :
Generalized Zeta Function (3)
The ordinary zeta function is a special case:
In certain cases, FunctionExpand gives formulas in terms of other functions:
Indefinite integral of the generalized zeta function:
Possible Issues (4)
Real and imaginary parts can have very different scales:
Evaluating the imaginary part accurately requires higher internal precision:
Machine-number inputs can give high‐precision results:
Giving 0 as an argument does not define the precision required:
Including an accuracy specification gives enough information:
In TraditionalForm , ζ is not automatically interpreted as the zeta function:
Neat Examples (2)
Play the real part of the zeta function on the critical line as a sound:
Animate the zeta function on the critical line:
See Also
PolyLog HarmonicNumber LerchPhi HurwitzZeta RiemannSiegelZ ZetaZero StieltjesGamma Glaisher PrimePi DirichletL RamanujanTauL PrimeZetaP DirichletTransform
Function Repository: ZetaSimplify
History
Introduced in 1988 (1.0) | Updated in 1999 (4.0) ▪ 2000 (4.1) ▪ 2002 (4.2) ▪ 2021 (13.0) ▪ 2022 (13.1)
Text
Wolfram Research (1988), Zeta, Wolfram Language function, https://reference.wolfram.com/language/ref/Zeta.html (updated 2022).
CMS
Wolfram Language. 1988. "Zeta." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2022. https://reference.wolfram.com/language/ref/Zeta.html.
APA
Wolfram Language. (1988). Zeta. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Zeta.html
BibTeX
@misc{reference.wolfram_2025_zeta, author="Wolfram Research", title="{Zeta}", year="2022", howpublished="\url{https://reference.wolfram.com/language/ref/Zeta.html}", note=[Accessed: 06-December-2025]}
BibLaTeX
@online{reference.wolfram_2025_zeta, organization={Wolfram Research}, title={Zeta}, year={2022}, url={https://reference.wolfram.com/language/ref/Zeta.html}, note=[Accessed: 06-December-2025]}