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


  • 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.


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Basic Examples  (2)

Evaluate to high precision:

Plot over a subset of the reals:

Scope  (12)

Numerical Evaluation  (4)

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:

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,


Wolfram Research (2008), PrimeZetaP, Wolfram Language function,


Wolfram Language. 2008. "PrimeZetaP." Wolfram Language & System Documentation Center. Wolfram Research.


Wolfram Language. (2008). PrimeZetaP. Wolfram Language & System Documentation Center. Retrieved from


@misc{reference.wolfram_2024_primezetap, author="Wolfram Research", title="{PrimeZetaP}", year="2008", howpublished="\url{}", note=[Accessed: 12-June-2024 ]}


@online{reference.wolfram_2024_primezetap, organization={Wolfram Research}, title={PrimeZetaP}, year={2008}, url={}, note=[Accessed: 12-June-2024 ]}