Details and Options
- Mathematical function, suitable for both symbolic and numerical manipulation.
- For , .
- , where any term with is excluded.
- For , the definition used is .
- 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. »
Examplesopen allclose all
Basic Examples (6)
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:
Numerical Evaluation (6)
The precision of the output tracks the precision of the input:
Evaluate efficiently at high precision:
Zeta can be used with Interval and CenteredInterval objects:
Specific Values (6)
Function Properties (12)
Real domain of Zeta:
The generalized zeta function has the same domain for all :
Zeta achieves all real values:
Zeta has the mirror property :
Zeta threads elementwise over lists and matrices:
Zeta is not an analytic function:
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:
has both singularity and discontinuity at :
is neither convex nor concave:
However, it is convex to the right of the singularity at :
Series Expansions (2)
Find the Taylor expansion using Series:
Plots of the first three approximations around :
General term in the series expansion using SeriesCoefficient:
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 what fraction of pairs of the first 100 integers are relatively prime:
Compare with a zeta function formula:
Test the Pustyl’nikov form of the Riemann hypothesis:
Plot real and imaginary parts in the vicinity of two very nearby zeros:
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)
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:
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:
Wolfram Research (1988), Zeta, Wolfram Language function, https://reference.wolfram.com/language/ref/Zeta.html (updated 2022).
Wolfram Language. 1988. "Zeta." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2022. https://reference.wolfram.com/language/ref/Zeta.html.
Wolfram Language. (1988). Zeta. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Zeta.html