# PrimeZetaP

PrimeZetaP[s]

gives prime zeta function .

# Details

• Mathematical function, suitable for both symbolic and numerical manipulation.
• 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(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:

## 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_2024_primezetap, author="Wolfram Research", title="{PrimeZetaP}", year="2008", howpublished="\url{https://reference.wolfram.com/language/ref/PrimeZetaP.html}", note=[Accessed: 12-June-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_primezetap, organization={Wolfram Research}, title={PrimeZetaP}, year={2008}, url={https://reference.wolfram.com/language/ref/PrimeZetaP.html}, note=[Accessed: 12-June-2024 ]}