# 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.
• RiemannXi can be used with Interval and CenteredInterval objects. »

# Examples

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## 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(25)

### Numerical Evaluation(6)

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:

Or compute average-case statistical intervals using Around:

Compute the elementwise values of an array:

Or compute the matrix RiemannXi function using MatrixFunction:

### 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(6)

RiemannXi has the mirror property :

RiemannXi is defined through the identity:

RiemannXi is neither non-increasing nor non-decreeing:

RiemannXi is not an injective function:

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:

## Applications(2)

Li's criterion states that the Riemann hypothesis is equivalent to the condition for all positive :

Generate and plot the first few values of :

Test the Pustylnikov form of the Riemann hypothesis, which states that all the even-order derivatives of the xi function are positive:

Wolfram Research (2014), RiemannXi, Wolfram Language function, https://reference.wolfram.com/language/ref/RiemannXi.html (updated 2022).

#### Text

Wolfram Research (2014), RiemannXi, Wolfram Language function, https://reference.wolfram.com/language/ref/RiemannXi.html (updated 2022).

#### CMS

Wolfram Language. 2014. "RiemannXi." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2022. https://reference.wolfram.com/language/ref/RiemannXi.html.

#### APA

Wolfram Language. (2014). RiemannXi. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/RiemannXi.html

#### BibTeX

@misc{reference.wolfram_2024_riemannxi, author="Wolfram Research", title="{RiemannXi}", year="2022", howpublished="\url{https://reference.wolfram.com/language/ref/RiemannXi.html}", note=[Accessed: 12-August-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_riemannxi, organization={Wolfram Research}, title={RiemannXi}, year={2022}, url={https://reference.wolfram.com/language/ref/RiemannXi.html}, note=[Accessed: 12-August-2024 ]}