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.
comment-Keiper's-lambda: >
Keiper CITE{Kei92} originally defined $\lambda_n$ with a different
$\lambda^{\text{Keiper}}_n = \frac{1}{n}\lambda_n$.
comment-Riemann-hypothesis: >
The Riemann hypothesis is equivalent to $\lambda_n\geq 0$ for all $n$.
formula-negative-indices: >
$\lambda_n = \lambda_{-n}$, if we extend the same definition
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
$\xi(s) = \frac{1}{2}s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)$
$\log(2\xi(1/z)) = \sum_{n=1}^\infty \frac{\lambda_{n}}{n} (1-z)^n$,
which is how Keiper originally defined the constants $\lambda^{\text{Keiper}}_n = \lambda_n/n$.
$\frac{d}{dz}\log\xi(\frac{1}{1-z}) = \sum_{n=0}^\infty \lambda_{n+1}z^n$.
$\lambda_1 = 1 + \gamma/2 - \log 2 - \frac{1}{2}\log \pi$,
where $\gamma$ is the Euler-Mascheroni constant.
$\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.
$\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)
formula-lambda-n-plus-1: >
$\lambda_{n+1} = \lambda_n +
\frac{1}{n!} \frac{d^n}{ds^n} \big( s^n \frac{\xi'(s)}{\xi(s)} \big) \big|_{s=1}$
Juan Arias de Reyna, "Asymptotics of Keiper-Li coefficients",
Functiones et Approximatio Commentarii Mathematici. 45(1) (2011), 7-21.
doi: 10.7169/facm/1317045228
M.W. Coffey, "Relations and positivity results for the
derivatives of the Riemann ξ function",
J. Comput. Appl. Math. 166 (2004), 525-534.
Donal F. Connon, "A recurrence relation for the Li/Keiper constants
in terms of the Stieltjes constants", (2009), 50 pages.
Fredrik Johansson, "Rigorous high-precision computation of the Hurwitz zeta function and its derivatives",
Numerical Algorithms. 69(2) (2015), 253-270.
doi: 10.1007/s11075-014-9893-1
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
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
Xian-Jin Li, "The Positivity of a Sequence of Numbers and the Riemann Hypothesis",
Journal of Number Theory 65 (1997), 325-333.
title: "Wikipedia: Riemann zeta function"
url: https://en.wikipedia.org/wiki/Riemann_zeta_function
title: "Wikipedia: Li's criterion"
url: https://en.wikipedia.org/wiki/Li%27s_criterion
Numbers: INPUT{numbers.yaml}