RiemannR

RiemannR[x]

gives the Riemann prime counting function .

Details

  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • For , the Riemann prime counting function is given by .
  • 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:

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:

The second largest root:

Introduced in 2008
 (7.0)