gives the Riemann zeta function .
gives the generalized Riemann zeta function .
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.
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:
Series expansion at a singular point:
Numerical Evaluation (5)
Evaluate to high precision:
The precision of the output tracks the precision of the input:
Complex number inputs:
Evaluate efficiently at high precision:
Specific Values (6)
Simple exact values are generated automatically:
Zeta[s,a] for symbolic :
Zeta[s,a] for symbolic :
Value at zero:
Limiting value at infinity:
Find a value of for which Zeta[s]=1.05:
Plot the Zeta function:
Plot the generalized Zeta function for various orders:
Plot the real part of the Zeta function:
Plot the imaginary part of the Zeta function:
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:
Function Identities and Simplifications (3)
Zeta is defined through the identity:
Sum involving the Zeta function:
Connection with the MoebiusMu function:
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:
Find what fraction of pairs of the first 100 integers are relatively prime:
Compare with a zeta function formula:
Define the xi function:
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:
Properties & Relations (8)
Riemann Zeta Function (5)
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:
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:
Introduced in 1988
Updated in 1999