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ID: INPUT{id.yaml}

Title: >
  Multiple zeta values of length 3

Definition: >
  This list contains values of the 
  multiple zeta function $\zeta(s_1,s_2,s_3)$ 
  for certain integers $s_1, s_2, s_3$.

Parameters:
  s1:
    display: $s_1$
    type: Z
    constraints: $s_1 \geq 2$

s2:
    display: $s_2$
    type: Z
    constraints: $s_2 \geq 1$

s3:
    display: $s_3$
    type: Z
    constraints: $s_3 \geq 1$
   
Comments:
  comment-multiple-zeta-function-definition: >
    For complex numbers $s_1,\ldots,s_k$ with 
    $\Re(s_1)+\ldots + \Re(s_i) > i$ (for all $i$), 
    the multiple zeta function is defined as
    $\zeta(s_1,\ldots,s_k) = \sum_{n_1 > n_2 > \ldots > n_k > 0} n^{-s_1}\cdots n^{-s_k}$.
  comment-multiple-zeta-values-definition: >
    The values of $\zeta(s_1,\ldots,s_k)$ 
    at positive integers $s_1,\ldots,s_k$ ($s_1 \geq 2$)
    are called multiple zeta values (MZVs) 
    of length $k$ and
    weight $s_1+\ldots + s_k$.
  
Formulas:

Programs: 
  program-sage:
    language: Sage
    code: |
      numbers = {s1: {s2: {s3: Multizeta(s3,s2,s1)
                      for s3 in [1..10]}
                      for s2 in [1..10]}
                      for s1 in [2..10]}

References:

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

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

Data properties:
  type: R
  complete: no
  
Display properties:
  number-header: $\zeta(s_1,s_2,s_3)$

Numbers: INPUT{numbers.yaml}

— preview —

Multiple zeta values of length 3

edit on github

Numbers

$s_1$

$s_2$

$s_3$

$\zeta(s_1,s_2,s_3)$

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

Definition

This list contains values of the multiple zeta function $\zeta(s_1,s_2,s_3)$ for certain integers $s_1, s_2, s_3$.

Parameters

$s_1$

— integer ($s_1 \geq 2$)

$s_2$

— integer ($s_2 \geq 1$)

$s_3$

— integer ($s_3 \geq 1$)

Comments

(1)

For complex numbers $s_1,\ldots,s_k$ with $\Re(s_1)+\ldots + \Re(s_i) > i$ (for all $i$), the multiple zeta function is defined as $\zeta(s_1,\ldots,s_k) = \sum_{n_1 > n_2 > \ldots > n_k > 0} n^{-s_1}\cdots n^{-s_k}$.

(2)

The values of $\zeta(s_1,\ldots,s_k)$ at positive integers $s_1,\ldots,s_k$ ($s_1 \geq 2$) are called multiple zeta values (MZVs) of length $k$ and weight $s_1+\ldots + s_k$.

Programs

(P1)

Sage

numbers = {s1: {s2: {s3: Multizeta(s3,s2,s1)
                for s3 in [1..10]}
                for s2 in [1..10]}
                for s1 in [2..10]}

Links

[1]

Wikipedia: Multiple zeta function

Data properties

Entries are of type: real number

Table is complete: no