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ID: INPUT{id.yaml}

Title: >
  Laurent series coefficients of the logarithmic derivative 
  of the completed Riemann zeta function $\xi(s)$ at $s=1$

Definition: >
  This list contains the coefficients of the Laurent series expansion
  $\xi'(s)/\xi(s) = \sum_{n=0}^\infty (-1)^n \sigma_{n+1} (s-1)^{n}$,
  where $\xi(s) = \frac{1}{2}s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)$
  is the completed Riemann zeta function CITE{WikiXi}.
  We additionally list $\sigma_n/n$, 
  the coefficients in CITE{formula-log-xi-at-1} and CITE{formula-log-xi-at-0}.
  
Parameters:
  expression:
    title: Transformation of constant
    type: Symbolic
    show-in-parameter-list: no

n:
    type: Z
    constraints: $n \geq 1$
  
Comments:
    
Formulas:
  formula-zeta-zero-moments: >
    For $n\geq 2$, 
    $\sigma_n = \sum_\rho \rho^{-n}$,
    where the sum is over the non-tirival zeros of $\zeta(s)$. CITE{Leh88}

formula-sigma-1: >
    $\sigma_1 = \lambda_1$, where $\lambda_1$ is the first
    Keipler-Li coefficient.

formula-log-xi-at-1: >
    $\log \xi(s) = - \log 2 - \sum_{n=1}^\infty (-1)^n \frac{\sigma_n}{n} (s-1)^n$.

formula-log-deriv-xi-at-0: >
    $\xi'(s)/\xi(s) = - \sum_{n=0}^\infty \sigma_{n+1} s^{n}$.

formula-log-xi-at-0: >
    $\log \xi(s) = - \log 2 - \sum_{n=1}^\infty \frac{\sigma_n}{n} s^n$.

formula-lambda: >
    $\lambda_n = - \sum_{k=1}^n (-1)^k \binom{n}{k} \sigma_k$. CITE{Cof05}

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.

Cof05:
    bib: >
      Mark W. Coffey, "Toward Verification of the Riemann Hypothesis: 
      Application of the Li Criterion",
      Mathematical Physics, Analysis and Geometry 8 (2005), 211-255.

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

Leh88: 
    bib: >
      D. H. Lehmer, "The sum of like powers of the zeros of the Riemann zeta function",
      Math. Comput. 50, 265-273 (1988).

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

Similar tables:
  
Keywords:
- zeta zeros
  
Tags:
- Taylor series
- L-function
- moments

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 completed Riemann zeta function $\xi(s)$ at $s=1$

edit on github

Numbers

$n$

$n$^th coefficient

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

Definition

This list contains the coefficients of the Laurent series expansion $\xi'(s)/\xi(s) = \sum_{n=0}^\infty (-1)^n \sigma_{n+1} (s-1)^{n}$, where $\xi(s) = \frac{1}{2}s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)$ is the completed Riemann zeta function [7]. We additionally list $\sigma_n/n$, the coefficients in (3) and (5).

Parameters

$n$

— integer ($n \geq 1$)

Formulas

(1)

For $n\geq 2$, $\sigma_n = \sum_\rho \rho^{-n}$, where the sum is over the non-tirival zeros of $\zeta(s)$. [5]

(2)

$\sigma_1 = \lambda_1$, where $\lambda_1$ is the first Keipler-Li coefficient.

(3)

$\log \xi(s) = - \log 2 - \sum_{n=1}^\infty (-1)^n \frac{\sigma_n}{n} (s-1)^n$.

(4)

$\xi'(s)/\xi(s) = - \sum_{n=0}^\infty \sigma_{n+1} s^{n}$.

(5)

$\log \xi(s) = - \log 2 - \sum_{n=1}^\infty \frac{\sigma_n}{n} s^n$.

(6)

$\lambda_n = - \sum_{k=1}^n (-1)^k \binom{n}{k} \sigma_k$. [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]

Mark W. Coffey, "Toward Verification of the Riemann Hypothesis: Application of the Li Criterion", Mathematical Physics, Analysis and Geometry 8 (2005), 211-255.

[3]

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

[4]

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

[5]

D. H. Lehmer, "The sum of like powers of the zeros of the Riemann zeta function", Math. Comput. 50, 265-273 (1988).

Links

[6]

Wikipedia: Riemann zeta function

[7]

Wikipedia: Riemann Xi function

Data properties

Entries are of type: real number

Table is complete: no