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RiemannR[x]

gives the Riemann prime counting function TemplateBox[{x}, RiemannR].

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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

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Basic Examples  (2)

Evaluate numerically:

Wolfram Language code: RiemannR[1000.]
Wolfram Language code: PrimePi[1000]

Compare the behavior of RiemannR with the prime counting function TemplateBox[{x}, PrimePi]:

Wolfram Language code: Plot[{RiemannR[x], PrimePi[x]}, {x, 1, 50}, PlotLegends -> "Expressions"]

Scope  (6)

Evaluate for complex arguments:

Wolfram Language code: RiemannR[1.5 + I]

Evaluate to high precision:

Wolfram Language code: N[RiemannR[1 / 2], 50]

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

Wolfram Language code: RiemannR[0.500000000000000000000000000000000000000]

Simple exact values are generated automatically:

Wolfram Language code: RiemannR[1]

RiemannR threads element-wise over lists:

Wolfram Language code: RiemannR[{2.0, 3.0, 4.0}]

TraditionalForm formatting:

Wolfram Language code: RiemannR[x]//TraditionalForm

Applications  (1)

The behavior of RiemannR near the origin:

Wolfram Language code: LogPlot[Abs[RiemannR[10 ^ x]], {x, -25000, 2}, Frame -> True, PlotStyle -> Thickness[0.006], Axes -> False, MaxRecursion -> 4, WorkingPrecision -> 20]//Quiet

The largest root of the Riemann prime counting function, which solves a problem originally posed by Waldvogel:

Wolfram Language code: lroot = FindRoot[RiemannR[10 ^ t], {t, -14000}, WorkingPrecision -> 20]
Wolfram Language code: 10 ^ t /. lroot
Wolfram Language code: ListLinePlot[N[Table[{t, RiemannR[10 ^ t]}, {t, -15500, -14500, 50}], 20], Epilog -> {Red, PointSize[0.05], Point[{t /. lroot, 0}]}]//Quiet

The second largest root:

Wolfram Language code: 10 ^ t /. FindRoot[RiemannR[10 ^ t], {t, -15300}, WorkingPrecision -> 20]
Wolfram Research (2008), RiemannR, Wolfram Language function, https://reference.wolfram.com/language/ref/RiemannR.html.

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_2026_riemannr, author="Wolfram Research", title="{RiemannR}", year="2008", howpublished="\url{https://reference.wolfram.com/language/ref/RiemannR.html}", note=[Accessed: 13-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_riemannr, organization={Wolfram Research}, title={RiemannR}, year={2008}, url={https://reference.wolfram.com/language/ref/RiemannR.html}, note=[Accessed: 13-June-2026]}