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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}

xxxxxxxxxx
 
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

edit on github

Numbers

n

λ_{n}

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

Definition

The Keiper-Li coefficients

λ_{n}

n \geq 0

, are defined as

λ_{n} = \sum_{ρ} (1 - (1 - \frac{1}{ρ})^{n})

, where the sum is interpreted as

lim_{T \to \infty} \sum_{ρ : | ρ | \leq T}

over the set of non-trivial zeros

ρ

of the Riemann zeta function.

Parameters

n

— integer (

n \geq 0

)

Formulas

(1)

λ_{n} = λ_{- n}

, if we extend the same definition to negative indices.

(2)

λ_{n} = \frac{1}{(n - 1)!} \frac{d^{n}}{d s^{n}} (s^{n - 1} \log ξ (s)) |_{s = 1}

for

n \geq 1

and

λ_{0} = 0

, where

ξ (s)

is the Riemann

ξ

function

ξ (s) = \frac{1}{2} s (s - 1) π^{- s / 2} Γ (s / 2) ζ (s)

(3)

\log (2 ξ (1 / z)) = \sum_{n = 1}^{\infty} \frac{λ_{n}}{n} (1 - z)^{n}

, which is how Keiper originally defined the constants

λ_{n}^{Keiper} = λ_{n} / n

(4)

\frac{d}{d z} \log ξ (\frac{1}{1 - z}) = \sum_{n = 0}^{\infty} λ_{n + 1} z^{n}

(5)

λ_{1} = 1 + γ / 2 - \log 2 - \frac{1}{2} \log π

, where

γ

is the Euler-Mascheroni constant.

(6)

λ_{2} = 1 + γ - γ^{2} + π^{2} / 8

- 2 \log 2 - \log π - 2 γ_{1}

, where

γ_{1}

is the first Stieltjes constant.

(7)

λ_{3} = \frac{1}{2} (2 + \frac{3}{4} π^{2} - 6 \log 2 - 3 \log π - 12 γ_{1} + γ (3 + 2 (γ - 3) γ + 6 γ_{1}) + 3 γ_{2} - \frac{7}{4} ζ (3))

(8)

λ_{n + 1} = λ_{n} + \frac{1}{n!} \frac{d^{n}}{d s^{n}} (s^{n} \frac{ξ^{'} (s)}{ξ (s)}) |_{s = 1}

for

n \geq 0

Comments

(9)

Keiper [5] originally defined

λ_{n}

with a different normalization:

λ_{n}^{Keiper} = \frac{1}{n} λ_{n}

(10)

The Riemann hypothesis is equivalent to

λ_{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

ξ

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