# RiemannXi

RiemannXi[s]

gives the Riemann xi function .

# Details • Mathematical function, suitable for symbolic and numeric manipulations.
• .
• For certain special arguments, RiemannXi automatically evaluates to exact values.
• RiemannXi is an entire function with no branch cut discontinuities.
• RiemannXi can be evaluated to arbitrary numerical precision.
• RiemannXi automatically threads over lists.

# Examples

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(22)

### Numerical Evaluation(4)

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:

### Specific Values(4)

Simple exact values are generated automatically:

Value at zero:

Evaluate symbolically:

Find the minimum of RiemannXi[x]:

### Visualization(2)

Plot the RiemannXi:

Plot the real part of the RiemannXi function:

Plot the imaginary part of the RiemannXi function:

### Function Properties(5)

Real domain of RiemannXi:

Complex domain:

RiemannXi has the mirror property :

RiemannXi is defined through the identity:

TraditionalForm formatting, while avoiding the evaluation:

### Differentiation(3)

First derivative with respect to :

Higher derivatives with respect to :

Plot the higher derivatives with respect to :

Formula for the  derivative with respect to :

### Series Expansions(4)

Find the Taylor expansion using Series:

Plots of the first three approximations around :

Find the series expansion at Infinity:

Find the series expansion for an arbitrary symbolic direction :

Taylor expansion at a generic point:

Introduced in 2014
(10.0)