RiemannR [x]
gives the Riemann prime counting function TemplateBox[{x}, RiemannR].
RiemannR
RiemannR [x]
gives the Riemann prime counting function TemplateBox[{x}, RiemannR].
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.
- 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)
Evaluate numerically:
Compare the behavior of RiemannR with the prime counting function TemplateBox[{x}, PrimePi]:
Scope (6)
Evaluate for complex arguments:
Evaluate to high precision:
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:
The second largest root:
See Also
Related Guides
Related Links
History
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
BibTeX
@misc{reference.wolfram_2025_riemannr, author="Wolfram Research", title="{RiemannR}", year="2008", howpublished="\url{https://reference.wolfram.com/language/ref/RiemannR.html}", note=[Accessed: 17-November-2025]}
BibLaTeX
@online{reference.wolfram_2025_riemannr, organization={Wolfram Research}, title={RiemannR}, year={2008}, url={https://reference.wolfram.com/language/ref/RiemannR.html}, note=[Accessed: 17-November-2025]}