![TemplateBox[{x}, RiemannR]](Files/RiemannR.en/8.png "TemplateBox[{x}, RiemannR]"){kind=small}
RiemannR
RiemannR[x]
gives the Riemann prime counting function ![TemplateBox[{x}, RiemannR]](Files/RiemannR.en/1.png "TemplateBox[{x}, RiemannR]"){kind=small}
.
Details
- Mathematical function, suitable for both symbolic and numerical manipulation.
- For
, the Riemann prime counting function is given by ![TemplateBox[{x}, RiemannR]=sum_n^inftyTemplateBox[{n}, MoebiusMu] TemplateBox[{{x, ^, {(, {1, /, n}, )}}}, LogIntegral]/n](Files/RiemannR.en/3.png "TemplateBox[{x}, RiemannR]=sum_n^inftyTemplateBox[{n}, MoebiusMu] TemplateBox[{{x, ^, {(, {1, /, n}, )}}}, LogIntegral]/n")
. - RiemannR[z] has a branch cut discontinuity in the complex z plane running from
to 
. - RiemannR can be evaluated to arbitrary numerical precision.
- RiemannR automatically threads over lists.
Examples
open all close allBasic Examples (2)
![TemplateBox[{x}, PrimePi]](Files/RiemannR.en/6.png "TemplateBox[{x}, PrimePi]"){kind=small}
Scope (6)
Evaluate for complex arguments:
The precision of the output tracks the precision of the input:
Simple exact values are generated automatically:
RiemannR threads element-wise over lists:
TraditionalForm formatting:
Applications (1)
The behavior of RiemannR near the origin:
The largest root of the Riemann prime counting function, which solves a problem originally posed by Waldvogel:
Text
Wolfram Research (2008), RiemannR, Wolfram Language function, https://reference.wolfram.com/language/ref/RiemannR.html.
CMS
Wolfram Language. 2008. "RiemannR." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/RiemannR.html.
APA
Wolfram Language. (2008). RiemannR. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/RiemannR.html