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ID: INPUT{id.yaml}

Title: >
  Stieltjes constants

Definition: >
  The Stieltjes constants $\gamma_n$, $n\geq 0$, can be defined via the
  Laurent series expansion of the Riemann zeta function at $s=1$:
  $\zeta(s) = \frac{1}{s-1} + 
   \sum_{n=0}^{\infty} \frac{(-1)^n}{n!}\gamma_n(s-1)^n$. 
  
Parameters:
  n:
    type: Z
    constraints: $n \neq 0$
  
Comments:
  comment-Riemann-zeta-definition: >
    The Riemann zeta function $\zeta(s)$
    is the meromorphic continuation of the Dirichlet series 
    $\sum_{n=0}^\infty n^{-s}, \Re(s) > 1$.

Formulas:
  formula-as-limit: >
    $\gamma_n = \lim_{m\to\infty} \sum_{k=1}^m 
        \big(\frac{\log^n k}{k} - \frac{\log^{n+1} m}{n+1}\big)$.

Programs: 
  program-sage:
    language: Sage
    code: |
      numbers = {n: stieltjes(n) for n in [0..100]}

References:
  Kei92:
    bib: >
      J. B. Keiper, "Power series expansions of Riemann's $\xi$ function",
      Math. Comp. 58 (1992), 765-773.
    doi: 10.1090/S0025-5718-1992-1122072-5
    MR: 1122072

Links:
  Wiki:
    title: "Wikipedia: Stieltjes constants"
    url: https://en.wikipedia.org/wiki/Stieltjes_constants

Similar tables:
  
Keywords:
  
Tags:
- Taylor series
- L-function

Data properties:
  type: R
  complete: no
  
Display properties:
  number-header: >
    $\gamma_n$

Numbers: INPUT{numbers.yaml}

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Stieltjes constants

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Numbers

$n$

$\gamma_n$

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

Definition

The Stieltjes constants $\gamma_n$, $n\geq 0$, can be defined via the Laurent series expansion of the Riemann zeta function at $s=1$: $\zeta(s) = \frac{1}{s-1} +
\sum_{n=0}^{\infty} \frac{(-1)^n}{n!}\gamma_n(s-1)^n$.

Parameters

$n$

— integer ($n \neq 0$)

Formulas

(1)

$\gamma_n = \lim_{m\to\infty} \sum_{k=1}^m
\big(\frac{\log^n k}{k} - \frac{\log^{n+1} m}{n+1}\big)$.

Comments

(2)

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

Programs

(P1)

Sage

numbers = {n: stieltjes(n) for n in [0..100]}

References

[1]

J. B. Keiper, "Power series expansions of Riemann's $\xi$ function", Math. Comp. 58 (1992), 765-773. (doi)

Links

[2]

Wikipedia: Stieltjes constants

Data properties

Entries are of type: real number

Table is complete: no