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collection.yaml
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collection.yaml
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ID: INPUT{id.yaml}
Title: >
Values of the prime zeta function $P(s)$ 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}