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ID: INPUT{id.yaml}

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

Definition: >
  The list contains values $\zeta(s)$ of the Riemann zeta function
  at rational numbers $s$.

Parameters:
  s:
    type: Q
    constraints: $s \neq 1$
   
Comments:
  comment-Riemann-zeta-definition: >
    The Riemann zeta function $\zeta(s)$
    is the meromorphic continuation of the Dirichlet series 
    $\sum_{n=0}^\infty n^{-s}, \Re(s) > 1$.
  
  comment-pole: >
    At $s=1$, $\zeta$ has a simple pole.
  
Formulas:
  formula-values-at-non-positive-integers: >
    $\zeta(-n) = (-1)^n\frac{B_n}{n+1}$ for integers $n\geq 0$, 
    where $B_n$ is the $n$'th HREF{T13}[Bernoulli number] 
    with sign convention $B_1=-1/2$.
    
  formula-values-at-positive-even-integers: >
    $\zeta(2n) = \frac{(-1)^{n+1}B_{2n}(2\pi)^{2n}}{2(2n)!}$ 
    for positive integers $n$.

Programs: 
  program-sage:
    language: Sage
    code: |
      numbers = {a/b: zeta(a/b) 
                      for b,a in cartesian_product(([1..20],[-40..40])) 
                      if gcd(a,b) == 1 and a/b != 1}

References:

Links:
  Wiki:
    title: "Wikipedia: Riemann zeta function"
    url: https://en.wikipedia.org/wiki/Riemann_zeta_function

Similar tables:
  
Keywords:
  
Tags:
- special values
- L-function

Data properties:
  type: R
  complete: no
  
Display properties:
  number-header: $\zeta(s)$

Numbers: INPUT{numbers.yaml}

— preview —

Values of the Riemann zeta function at rational numbers

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Numbers

$s$

$\zeta(s)$

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

Definition

The list contains values $\zeta(s)$ of the Riemann zeta function at rational numbers $s$.

Parameters

$s$

— rational number ($s \neq 1$)

Formulas

(1)

$\zeta(-n) = (-1)^n\frac{B_n}{n+1}$ for integers $n\geq 0$, where $B_n$ is the $n$'th Bernoulli number with sign convention $B_1=-1/2$.

(2)

$\zeta(2n) = \frac{(-1)^{n+1}B_{2n}(2\pi)^{2n}}{2(2n)!}$ for positive integers $n$.

Comments

(3)

The Riemann zeta function $\zeta(s)$ is the meromorphic continuation of the Dirichlet series $\sum_{n=0}^\infty n^{-s}, \Re(s) > 1$.

(4)

At $s=1$, $\zeta$ has a simple pole.

Programs

(P1)

Sage

numbers = {a/b: zeta(a/b) 
                for b,a in cartesian_product(([1..20],[-40..40])) 
                if gcd(a,b) == 1 and a/b != 1}

Links

[1]

Wikipedia: Riemann zeta function

Data properties

Entries are of type: real number

Table is complete: no