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PrimeZetaP[s]

gives prime zeta function TemplateBox[{s}, PrimeZetaP].

Details
Details and Options Details and Options
Examples  
Basic Examples  
Scope  
Numerical Evaluation  
Specific Values  
Visualization  
Function Properties  
Applications  
Neat Examples  
See Also
Related Guides
History
Cite this Page

PrimeZetaP

PrimeZetaP[s]

gives prime zeta function TemplateBox[{s}, PrimeZetaP].

Details

  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • TemplateBox[{s}, PrimeZetaP] is defined by for and by analytic continuation for .
  • PrimeZetaP can be evaluated to arbitrary numerical precision.
  • PrimeZetaP automatically threads over lists. »

Examples

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Basic Examples  (2)

Evaluate to high precision:

Plot over a subset of the reals:

Scope  (14)

Numerical Evaluation  (6)

Evaluate numerically:

Evaluate to high precision:

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

Complex number input:

Evaluate efficiently at high precision:

Compute the elementwise values of an array:

Or compute the matrix PrimeZetaP function using MatrixFunction:

Compute average-case statistical intervals using Around:

Specific Values  (3)

Simple exact values are generated automatically:

Some singular points of PrimeZetaP:

Find a value of s for which Re[PrimeZetaP[s]]=-2:

Visualization  (2)

Plot the real and imaginary parts of PrimeZetaP function:

Plot the real part of PrimeZetaP function:

Function Properties  (3)

Real domain of PrimeZetaP:

Complex domain:

PrimeZetaP threads element-wise over lists:

TraditionalForm formatting:

Applications  (2)

Compute Artin's constant:

Compute an approximation to the first Mertens constant:

Neat Examples  (1)

A twisted curve in the complex plane, based on the prime zeta function:

Wolfram Research (2008), PrimeZetaP, Wolfram Language function, https://reference.wolfram.com/language/ref/PrimeZetaP.html.

Text

Wolfram Research (2008), PrimeZetaP, Wolfram Language function, https://reference.wolfram.com/language/ref/PrimeZetaP.html.

CMS

Wolfram Language. 2008. "PrimeZetaP." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/PrimeZetaP.html.

APA

Wolfram Language. (2008). PrimeZetaP. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/PrimeZetaP.html

BibTeX

@misc{reference.wolfram_2025_primezetap, author="Wolfram Research", title="{PrimeZetaP}", year="2008", howpublished="\url{https://reference.wolfram.com/language/ref/PrimeZetaP.html}", note=[Accessed: 28-August-2025]}

BibLaTeX

@online{reference.wolfram_2025_primezetap, organization={Wolfram Research}, title={PrimeZetaP}, year={2008}, url={https://reference.wolfram.com/language/ref/PrimeZetaP.html}, note=[Accessed: 28-August-2025]}