RiemannXi
✖
RiemannXi
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
open allclose allBasic Examples (6)Summary of the most common use cases

https://wolfram.com/xid/0i1p7c4k7-g0m

Plot over a subset of the reals:

https://wolfram.com/xid/0i1p7c4k7-iafz3q

Plot over a subset of the complexes:

https://wolfram.com/xid/0i1p7c4k7-kiedlx

Series expansion at the origin:

https://wolfram.com/xid/0i1p7c4k7-fdkkja

Series expansion at Infinity:

https://wolfram.com/xid/0i1p7c4k7-20imb

Series expansion at a singular point:

https://wolfram.com/xid/0i1p7c4k7-d2klx1

Scope (25)Survey of the scope of standard use cases
Numerical Evaluation (6)

https://wolfram.com/xid/0i1p7c4k7-l274ju


https://wolfram.com/xid/0i1p7c4k7-cksbl4


https://wolfram.com/xid/0i1p7c4k7-b0wt9


https://wolfram.com/xid/0i1p7c4k7-zn1q5

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

https://wolfram.com/xid/0i1p7c4k7-y7k4a


https://wolfram.com/xid/0i1p7c4k7-hfml09


https://wolfram.com/xid/0i1p7c4k7-dm5qi7

Evaluate efficiently at high precision:

https://wolfram.com/xid/0i1p7c4k7-di5gcr


https://wolfram.com/xid/0i1p7c4k7-bq2c6r

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

https://wolfram.com/xid/0i1p7c4k7-e6ahxc


https://wolfram.com/xid/0i1p7c4k7-cmdnbi

Or compute average-case statistical intervals using Around:

https://wolfram.com/xid/0i1p7c4k7-cw18bq

Compute the elementwise values of an array:

https://wolfram.com/xid/0i1p7c4k7-thgd2

Or compute the matrix RiemannXi function using MatrixFunction:

https://wolfram.com/xid/0i1p7c4k7-o5jpo

Specific Values (4)
Simple exact values are generated automatically:

https://wolfram.com/xid/0i1p7c4k7-nww7l


https://wolfram.com/xid/0i1p7c4k7-bq4ikv


https://wolfram.com/xid/0i1p7c4k7-bmqd0y


https://wolfram.com/xid/0i1p7c4k7-cs023

Find the minimum of RiemannXi[x]:

https://wolfram.com/xid/0i1p7c4k7-f2hrld


https://wolfram.com/xid/0i1p7c4k7-k6sdsx

Visualization (2)
Function Properties (6)
RiemannXi has the mirror property :

https://wolfram.com/xid/0i1p7c4k7-heoddu

RiemannXi is defined through the identity:

https://wolfram.com/xid/0i1p7c4k7-b5js3b

RiemannXi threads element‐wise over lists:

https://wolfram.com/xid/0i1p7c4k7-f7760m

RiemannXi is neither non-increasing nor non-decreeing:

https://wolfram.com/xid/0i1p7c4k7-g6kynf

RiemannXi is not an injective function:

https://wolfram.com/xid/0i1p7c4k7-gn6dxi


https://wolfram.com/xid/0i1p7c4k7-pi1dox

TraditionalForm formatting, while avoiding the evaluation:

https://wolfram.com/xid/0i1p7c4k7-f4gt

Differentiation (3)
First derivative with respect to :

https://wolfram.com/xid/0i1p7c4k7-krpoah

Higher derivatives with respect to :

https://wolfram.com/xid/0i1p7c4k7-z33jv

Plot the higher derivatives with respect to :

https://wolfram.com/xid/0i1p7c4k7-fxwmfc

Formula for the derivative with respect to
:

https://wolfram.com/xid/0i1p7c4k7-cb5zgj

Series Expansions (4)
Find the Taylor expansion using Series:

https://wolfram.com/xid/0i1p7c4k7-ewr1h8

Plots of the first three approximations around :

https://wolfram.com/xid/0i1p7c4k7-binhar

Find the series expansion at Infinity:

https://wolfram.com/xid/0i1p7c4k7-syq

Find the series expansion for an arbitrary symbolic direction :

https://wolfram.com/xid/0i1p7c4k7-t5t

Taylor expansion at a generic point:

https://wolfram.com/xid/0i1p7c4k7-jwxla7

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
:

https://wolfram.com/xid/0i1p7c4k7-eer9ze
Generate and plot the first few values of :

https://wolfram.com/xid/0i1p7c4k7-benf5e


https://wolfram.com/xid/0i1p7c4k7-ccyj8q

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

https://wolfram.com/xid/0i1p7c4k7-cjqpbb

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
]}
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
]}