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ID: INPUT{id.yaml}

Title: >
  Keiper-Li coefficients

Definition: >
  The Keiper-Li coefficients $\lambda_n$, $n\geq 0$, are defined as
  $\lambda_n = \sum_\rho \big(1-(1-\frac{1}{\rho})^n\big)$,
  where the sum is interpreted as 
  $\lim_{T\to\infty} \sum_{\rho:|\rho|\leq T}$
  over the set of non-trivial zeros $\rho$ of the Riemann zeta function.
  
Parameters:
  n:
    type: Z
    constraints: $n \geq 0$
  
Comments:
  comment-Keiper's-lambda: >
    Keiper CITE{Kei92} originally defined $\lambda_n$ with a different
    normalization:
    $\lambda^{\text{Keiper}}_n = \frac{1}{n}\lambda_n$.

comment-Riemann-hypothesis: >
    The Riemann hypothesis is equivalent to $\lambda_n\geq 0$ for all $n$.
    This is Li's criterion.
    
Formulas:
  formula-negative-indices: >
    $\lambda_n = \lambda_{-n}$, if we extend the same definition 
    to negative indices.
  
  formula-original-definition: >
    $\lambda_n = \frac{1}{(n-1)!} \frac{d^n}{ds^n}\big(s^{n-1}\log\xi(s)\big)|_{s=1}$ 
    for $n\geq 1$ and $\lambda_0 = 0$, 
    where $\xi(s)$ is the Riemann $\xi$ function
    $\xi(s) = \frac{1}{2}s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)$

formula-taylor-1: >
    $\log(2\xi(1/z)) = \sum_{n=1}^\infty \frac{\lambda_{n}}{n} (1-z)^n$,
    which is how Keiper originally defined the constants $\lambda^{\text{Keiper}}_n = \lambda_n/n$.
    
  formula-taylor-2: >
    $\frac{d}{dz}\log\xi(\frac{1}{1-z}) = \sum_{n=0}^\infty \lambda_{n+1}z^n$.
    
  formula-lambda-1: >
    $\lambda_1 = 1 + \gamma/2 - \log 2 - \frac{1}{2}\log \pi$, 
    where $\gamma$ is the Euler-Mascheroni constant.
    
  formula-lambda-2: >
    $\lambda_2 = 1 +\gamma - \gamma^2 + \pi^2/8 $ $- 2\log 2 - \log\pi - 2\gamma_1$, 
    where $\gamma_1$ is the first Stieltjes constant.
  
  formula-lambda-3: >
    $\lambda_3 = \frac{1}{2}\big(
    2 + \frac{3}{4}\pi^2 - 6\log 2 - 3\log\pi - 12\gamma_1 + 
    \gamma(3 + 2(\gamma - 3)\gamma + 6\gamma_1) + 3\gamma_2 - \frac{7}{4}\zeta(3)
    \big)$.
    
  formula-lambda-n-plus-1: >
    $\lambda_{n+1} = \lambda_n + 
    \frac{1}{n!} \frac{d^n}{ds^n} \big( s^n \frac{\xi'(s)}{\xi(s)} \big) \big|_{s=1}$
    for $n\geq 0$.
  
Programs:

References:
  AdR11:
    bib: >
      Juan Arias de Reyna, "Asymptotics of Keiper-Li coefficients",
      Functiones et Approximatio Commentarii Mathematici. 45(1) (2011), 7-21.
    doi: 10.7169/facm/1317045228

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

Joh15:
    bib: >
      Fredrik Johansson, "Rigorous high-precision computation of the Hurwitz zeta function and its derivatives",
      Numerical Algorithms. 69(2) (2015), 253-270. 
    arXiv: 1309.2877
    doi: 10.1007/s11075-014-9893-1
      
  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

Lag07:
    bib: >
      Jeffrey C. Lagarias, "Li coefficients for automorphic L-functions",
      Ann. Inst. Fourier, Grenoble, Tome 57, no 5 (2007), 1689-1740.
    url: http://aif.cedram.org/item?id=AIF_2007__57_5_1689_0

Li97:
    bib: >
      Xian-Jin Li, "The Positivity of a Sequence of Numbers and the Riemann Hypothesis",
      Journal of Number Theory 65 (1997), 325-333.

Links:
  WikiZeta:
    title: "Wikipedia: Riemann zeta function"
    url: https://en.wikipedia.org/wiki/Riemann_zeta_function
  WikiLi:
    title: "Wikipedia: Li's criterion"
    url: https://en.wikipedia.org/wiki/Li%27s_criterion

Similar tables:
  
Keywords:
- RH
- Riemann hypothesis
  
Tags:
- Taylor series
- L-function

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

Numbers: INPUT{numbers.yaml}

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Keiper-Li coefficients

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Numbers

$n$

$\lambda_n$

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

Definition

The Keiper-Li coefficients $\lambda_n$, $n\geq 0$, are defined as $\lambda_n = \sum_\rho \big(1-(1-\frac{1}{\rho})^n\big)$, where the sum is interpreted as $\lim_{T\to\infty} \sum_{\rho:|\rho|\leq T}$ over the set of non-trivial zeros $\rho$ of the Riemann zeta function.

Parameters

$n$

— integer ($n \geq 0$)

Formulas

(1)

$\lambda_n = \lambda_{-n}$, if we extend the same definition to negative indices.

(2)

$\lambda_n = \frac{1}{(n-1)!} \frac{d^n}{ds^n}\big(s^{n-1}\log\xi(s)\big)|_{s=1}$ for $n\geq 1$ and $\lambda_0 = 0$, where $\xi(s)$ is the Riemann $\xi$ function $\xi(s) = \frac{1}{2}s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)$

(3)

$\log(2\xi(1/z)) = \sum_{n=1}^\infty \frac{\lambda_{n}}{n} (1-z)^n$, which is how Keiper originally defined the constants $\lambda^{\text{Keiper}}_n = \lambda_n/n$.

(4)

$\frac{d}{dz}\log\xi(\frac{1}{1-z}) = \sum_{n=0}^\infty \lambda_{n+1}z^n$.

(5)

$\lambda_1 = 1 + \gamma/2 - \log 2 - \frac{1}{2}\log \pi$, where $\gamma$ is the Euler-Mascheroni constant.

(6)

$\lambda_2 = 1 +\gamma - \gamma^2 + \pi^2/8 $ $- 2\log 2 - \log\pi - 2\gamma_1$, where $\gamma_1$ is the first Stieltjes constant.

(7)

$\lambda_3 = \frac{1}{2}\big( 2 + \frac{3}{4}\pi^2 - 6\log 2 - 3\log\pi - 12\gamma_1 + \gamma(3 + 2(\gamma - 3)\gamma + 6\gamma_1) + 3\gamma_2 - \frac{7}{4}\zeta(3) \big)$.

(8)

$\lambda_{n+1} = \lambda_n + \frac{1}{n!} \frac{d^n}{ds^n} \big( s^n \frac{\xi'(s)}{\xi(s)} \big) \big|_{s=1}$ for $n\geq 0$.

Comments

(9)

Keiper [5] originally defined $\lambda_n$ with a different normalization: $\lambda^{\text{Keiper}}_n = \frac{1}{n}\lambda_n$.

(10)

The Riemann hypothesis is equivalent to $\lambda_n\geq 0$ for all $n$. This is Li's criterion.

References

[1]

Juan Arias de Reyna, "Asymptotics of Keiper-Li coefficients", Functiones et Approximatio Commentarii Mathematici. 45(1) (2011), 7-21. (doi)

[2]

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

[3]

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

[4]

Fredrik Johansson, "Rigorous high-precision computation of the Hurwitz zeta function and its derivatives", Numerical Algorithms. 69(2) (2015), 253-270. (arXiv) (doi)

[5]

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

[6]

Jeffrey C. Lagarias, "Li coefficients for automorphic L-functions", Ann. Inst. Fourier, Grenoble, Tome 57, no 5 (2007), 1689-1740. http://aif.cedram.org/item?id=AIF_2007__57_5_1689_0

[7]

Xian-Jin Li, "The Positivity of a Sequence of Numbers and the Riemann Hypothesis", Journal of Number Theory 65 (1997), 325-333.

Links

[8]

Wikipedia: Riemann zeta function

[9]

Wikipedia: Li's criterion

Data properties

Entries are of type: real number

Table is complete: no