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ID: INPUT{id.yaml}

Title: >
  Values of the prime zeta function at rational numbers

Definition: >
  For $\Re(s)>1$, one defines $P(s) = \sum_{p\in \text{primes}} p^{-s}$,
  which can be analytically continued to $\Re(s)>0$.
  This list contains values $P(s)$ for certain rational numbers $s$.

Parameters:
  s:
    type: C
    constraints: $\Re(s) > 0$, $s$ not the inverse of a squarefree integer
   
Comments:
  comment-poles: >
    $P(s)$ has its poles at $1/n$, 
    where $n$ runs over all squarefree positive integers.
  
  
Formulas:

Programs: 
  program-sage:
    language: Sage
    code: |
      from mpmath import mp
      numbers = {a/b: mp.primezeta(a/b)
                      for b,a in cartesian_product(([1..20],[1..20])) 
                      if gcd(a,b) == 1 and a/b != 1 and
                      (a != 1 or not b.is_squarefree())}

References:

Links:
  mpmath: >
    HREF{https://mpmath.org/}[mpmath] (Python library)
  Wiki:
    title: "Wikipedia: Prime zeta function"
    url: https://en.wikipedia.org/wiki/Prime_zeta_function

Similar tables:
  
Keywords:
  
Tags:
- special values
- Dirichlet series

Data properties:
  type: R
  complete: no
  reliability: computed with mpmath CITE{mpmath}
  
Display properties:
  number-header: $P(s)$

Numbers: INPUT{numbers.yaml}

— preview —

Values of the prime zeta function at rational numbers

edit on github

Numbers

$s$

$P(s)$

INPUT{numbers.yaml} (not shown in preview)

Definition

For $\Re(s)>1$, one defines $P(s) = \sum_{p\in \text{primes}} p^{-s}$, which can be analytically continued to $\Re(s)>0$. This list contains values $P(s)$ for certain rational numbers $s$.

Parameters

$s$

— complex number ($\Re(s) > 0$, $s$ not the inverse of a squarefree integer)

Comments

(1)

$P(s)$ has its poles at $1/n$, where $n$ runs over all squarefree positive integers.

Programs

(P1)

Sage

from mpmath import mp
numbers = {a/b: mp.primezeta(a/b)
                for b,a in cartesian_product(([1..20],[1..20])) 
                if gcd(a,b) == 1 and a/b != 1 and
                (a != 1 or not b.is_squarefree())}

Links

[1]

mpmath (Python library)

[2]

Wikipedia: Prime zeta function

Data properties

Entries are of type: real number

Table is complete: no

Reliability: computed with mpmath [1]