WOLFRAM

RiemannXi
RiemannXi

gives the Riemann xi function TemplateBox[{s}, RiemannXi].

Details

  • Mathematical function, suitable for symbolic and numeric manipulations.
  • TemplateBox[{s}, RiemannXi]=1/2 s (s-1) pi^(-s/2) TemplateBox[{s}, Zeta] TemplateBox[{{s, /, 2}}, Gamma].
  • 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)Summary of the most common use cases

Evaluate numerically:

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Plot over a subset of the reals:

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Plot over a subset of the complexes:

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Series expansion at the origin:

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Series expansion at Infinity:

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Series expansion at a singular point:

Out[1]=1

Scope  (25)Survey of the scope of standard use cases

Numerical Evaluation  (6)

Evaluate numerically:

Out[2]=2
Out[1]=1

Evaluate to high precision:

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Out[2]=2

The precision of the output tracks the precision of the input:

Out[3]=3

Complex number inputs:

Out[1]=1
Out[2]=2

Evaluate efficiently at high precision:

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Out[2]=2

Compute worst-case guaranteed intervals using Interval and CenteredInterval objects:

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Out[2]=2

Or compute average-case statistical intervals using Around:

Out[3]=3

Compute the elementwise values of an array:

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Or compute the matrix RiemannXi function using MatrixFunction:

Out[2]=2

Specific Values  (4)

Simple exact values are generated automatically:

Out[1]=1
Out[2]=2

Value at zero:

Out[1]=1

Evaluate symbolically:

Out[1]=1

Find the minimum of RiemannXi[x]:

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Out[2]=2

Visualization  (2)

Plot the RiemannXi:

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Plot the real part of the RiemannXi function:

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Plot the imaginary part of the RiemannXi function:

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Function Properties  (6)

RiemannXi has the mirror property TemplateBox[{TemplateBox[{z}, Conjugate, SyntaxForm -> SuperscriptBox]}, RiemannXi]=TemplateBox[{TemplateBox[{z}, RiemannXi]}, Conjugate]:

Out[1]=1

RiemannXi is defined through the identity:

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RiemannXi threads elementwise over lists:

Out[1]=1

RiemannXi is neither non-increasing nor non-decreeing:

Out[1]=1

RiemannXi is not an injective function:

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Out[2]=2

TraditionalForm formatting, while avoiding the evaluation:

Differentiation  (3)

First derivative with respect to :

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Higher derivatives with respect to :

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Plot the higher derivatives with respect to :

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Formula for the ^(th) derivative with respect to :

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Series Expansions  (4)

Find the Taylor expansion using Series:

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Plots of the first three approximations around :

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Find the series expansion at Infinity:

Out[1]=1

Find the series expansion for an arbitrary symbolic direction :

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Taylor expansion at a generic point:

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Applications  (2)Sample problems that can be solved with this function

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

Generate and plot the first few values of :

Out[2]=2
Out[3]=3

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

Out[1]=1
Wolfram Research (2014), RiemannXi, Wolfram Language function, https://reference.wolfram.com/language/ref/RiemannXi.html (updated 2022).
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).

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.

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

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

BibTeX

@misc{reference.wolfram_2025_riemannxi, author="Wolfram Research", title="{RiemannXi}", year="2022", howpublished="\url{https://reference.wolfram.com/language/ref/RiemannXi.html}", note=[Accessed: 29-March-2025 ]}

@misc{reference.wolfram_2025_riemannxi, author="Wolfram Research", title="{RiemannXi}", year="2022", howpublished="\url{https://reference.wolfram.com/language/ref/RiemannXi.html}", note=[Accessed: 29-March-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_riemannxi, organization={Wolfram Research}, title={RiemannXi}, year={2022}, url={https://reference.wolfram.com/language/ref/RiemannXi.html}, note=[Accessed: 29-March-2025 ]}

@online{reference.wolfram_2025_riemannxi, organization={Wolfram Research}, title={RiemannXi}, year={2022}, url={https://reference.wolfram.com/language/ref/RiemannXi.html}, note=[Accessed: 29-March-2025 ]}