Preview for editing tables

In this editor you can enter or edit tables in YAML format, and you can get a preview of how numberdb would render it. You cannot save changes here. Once everything looks good, upload it to the git repository numberdb-data.

ID: INPUT{id.yaml}

Title: >
  Laurent series coefficients of the logarithmic derivative 
  of the Riemann zeta function at 1

Definition: >
  This list contains the coefficients $\eta_n$ of the Laurent series expansion
  $\zeta'(s)/\zeta(s) = - \sum_{n=-1}^\infty \eta_n (s-1)^n$.
  We additionally list $\eta_n/n$, 
  the coefficients in CITE{formula-log-zeta-1}.
  
Parameters:
  expression:
    title: Transformation of constant
    type: Symbolic
    show-in-parameter-list: no

n:
    type: Z
    constraints: $n \geq -1$
  
Comments:
  comment-radius-of-convergence: >
    The radius of convergence of the expansion of $\zeta'(s)/\zeta(s)$
    at $s=1$ is $3$, due to the trivial zero of $\zeta(s)$ at $s=-2$.
    
Formulas:
  formula-eta_0: >
    $\eta_0 = -\gamma$, where $\gamma=\gamma_0$ is the Euler-Mascheroni constant.

formula-recurrence: >
    $\eta_n = -(-1)^n \frac{n+1}{n!}\gamma_n + \sum_{k=0}^{n-1} \frac{(-1)^{n-k}}{(n-k-1)!}\eta_k \gamma_{n-k-1}$
    for $n\geq 0$, where $\gamma_n$ are the Stieltjes constants. CITE{Cof04}

formula-log-zeta-1: >
    $\log \zeta(s) = -\log(s-1) - \sum_{n=1}^\infty \frac{\eta_n}{n} (s-1)^n$.

Programs:

References:
  Cof04:
    bib: >
      M.W. Coffey, "Relations and positivity results for the 
      derivatives of the Riemann ξ function",
      J. Comput. Appl. Math. 166 (2004), 525-534.

Con09:
    bib: >
      Donal F. Connon, "A recurrence relation for the Li/Keiper constants 
      in terms of the Stieltjes constants", (2009), 50 pages.
    arXiv: 0902.1691

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

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

Data properties:
  type: R
  complete: no
  
Display properties:
  number-header: >
    $n$<sup>th</sup> coefficient

Numbers: INPUT{numbers.yaml}

— preview —

Laurent series coefficients of the logarithmic derivative of the Riemann zeta function at 1

edit on github

Numbers

$n$

$n$^th coefficient

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

Definition

This list contains the coefficients $\eta_n$ of the Laurent series expansion $\zeta'(s)/\zeta(s) = - \sum_{n=-1}^\infty \eta_n (s-1)^n$. We additionally list $\eta_n/n$, the coefficients in (3).

Parameters

$n$

— integer ($n \geq -1$)

Formulas

(1)

$\eta_0 = -\gamma$, where $\gamma=\gamma_0$ is the Euler-Mascheroni constant.

(2)

$\eta_n = -(-1)^n \frac{n+1}{n!}\gamma_n + \sum_{k=0}^{n-1} \frac{(-1)^{n-k}}{(n-k-1)!}\eta_k \gamma_{n-k-1}$ for $n\geq 0$, where $\gamma_n$ are the Stieltjes constants. [1]

(3)

$\log \zeta(s) = -\log(s-1) - \sum_{n=1}^\infty \frac{\eta_n}{n} (s-1)^n$.

Comments

(4)

The radius of convergence of the expansion of $\zeta'(s)/\zeta(s)$ at $s=1$ is $3$, due to the trivial zero of $\zeta(s)$ at $s=-2$.

References

[1]

M.W. Coffey, "Relations and positivity results for the derivatives of the Riemann ξ function", J. Comput. Appl. Math. 166 (2004), 525-534.

[2]

Donal F. Connon, "A recurrence relation for the Li/Keiper constants in terms of the Stieltjes constants", (2009), 50 pages. (arXiv)

[3]

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

Links

[4]

Wikipedia: Riemann zeta function

Data properties

Entries are of type: real number

Table is complete: no