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RiemannR
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RiemannR

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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 allclose all

Basic Examples  (2)Summary of the most common use cases

Evaluate numerically:

In[1]:=1
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https://wolfram.com/xid/0j45ozy5-b74byy
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0j45ozy5-haizjn
Out[2]=2

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

In[1]:=1
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https://wolfram.com/xid/0j45ozy5-hg6toh
Out[1]=1

Scope  (6)Survey of the scope of standard use cases

Evaluate for complex arguments:

In[1]:=1
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https://wolfram.com/xid/0j45ozy5-bk50jw
Out[1]=1

Evaluate to high precision:

In[1]:=1
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https://wolfram.com/xid/0j45ozy5-bpc23o
Out[1]=1

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

In[1]:=1
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https://wolfram.com/xid/0j45ozy5-zob0n
Out[1]=1

Simple exact values are generated automatically:

In[1]:=1
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https://wolfram.com/xid/0j45ozy5-exgtl
Out[1]=1

RiemannR threads element-wise over lists:

In[1]:=1
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https://wolfram.com/xid/0j45ozy5-i7bdrl
Out[1]=1

TraditionalForm formatting:

In[1]:=1
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https://wolfram.com/xid/0j45ozy5-ggmxww

Applications  (1)Sample problems that can be solved with this function

The behavior of RiemannR near the origin:

In[1]:=1
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https://wolfram.com/xid/0j45ozy5-hk2tof
Out[1]=1

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

In[2]:=2
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https://wolfram.com/xid/0j45ozy5-cjrofb
Out[2]=2
In[3]:=3
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https://wolfram.com/xid/0j45ozy5-galrx0
Out[3]=3
In[4]:=4
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https://wolfram.com/xid/0j45ozy5-lmkhsv
Out[4]=4

The second largest root:

In[5]:=5
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https://wolfram.com/xid/0j45ozy5-bi6gr
Out[5]=5
Wolfram Research (2008), RiemannR, Wolfram Language function, https://reference.wolfram.com/language/ref/RiemannR.html.
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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.

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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.

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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

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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: 29-March-2025 ]}

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@misc{reference.wolfram_2025_riemannr, author="Wolfram Research", title="{RiemannR}", year="2008", howpublished="\url{https://reference.wolfram.com/language/ref/RiemannR.html}", note=[Accessed: 29-March-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: 29-March-2025 ]}

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@online{reference.wolfram_2025_riemannr, organization={Wolfram Research}, title={RiemannR}, year={2008}, url={https://reference.wolfram.com/language/ref/RiemannR.html}, note=[Accessed: 29-March-2025 ]}