KeiperLiLambda

The n-th Keiper–Li coefficient $\lambda_n = \dfrac{1}{(n-1)!}\dfrac{d^n}{ds^n}\left[s^{n-1}\log\xi(s)\right]_{s=1}$, whose nonnegativity for every n is equivalent to the Riemann hypothesis.

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KeiperLiLambda(n)the n-th Keiper–Li coefficient, for a nonnegative integer n.

Typed signatures

Type	Package	Applies
(integer) -> number	analytic	always

Domain: Special functions

Details

• $\xi(s) = \tfrac12 s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)$, the completed (Riemann) xi function — entire, and sharing ζ's nontrivial zeros.
• $\lambda_0 = 0$ and $\lambda_1 = 1 + \gamma/2 - \tfrac12\ln(4\pi) \approx 0.02310$ are closed forms; both evaluate exactly under plain evaluation, not just under N().
• $n \geq 2$ is computed by Cauchy's differentiation formula — a contour integral of $\log\xi$ around $s=1$ — since repeated finite differences lose too much precision by n = 2 or 3.
• Numeric only for n ≥ 2: needs N() or a floating-point argument.
• Declines (leaves the call unevaluated) past n = 20, where float64 rounding in the contour sum starts costing real digits, and for any negative or non-integer n.
• Li's criterion: the Riemann hypothesis holds if and only if $\lambda_n \geq 0$ for every positive integer n.

Examples

17 more kept as data, checked against the oracles

Basic #

λ₀ = 0, exact under plain evaluation

live ↗

In$\mathrm{KeiperLiLambda}(0)$Out$0$

Known value · mpmath 1.3.0 (independent Cauchy contour-integral implementation)

λ₁'s closed form, exact under plain evaluation

live ↗

In$\mathrm{KeiperLiLambda}(1)$Out$1-\frac{\ln(4π)}{2}+\frac{γ}{2}$

N() forces the closed form to a decimal

live ↗

In$N(\mathrm{KeiperLiLambda}(1))$Out$0.023\,095\,708\,966\,121\,033\,813\,5$

Known value · mpmath 1.3.0 (independent Cauchy contour-integral implementation)

n ≥ 2 is numeric-only: the contour-integral evaluator

live ↗

In$N(\mathrm{KeiperLiLambda}(2))$Out$0.092\,345\,735\,228\,047\,94$

Known value · mpmath 1.3.0 (independent Cauchy contour-integral implementation)

live ↗

In$N(\mathrm{KeiperLiLambda}(3))$Out$0.207\,638\,920\,554\,332\,03$

Known value · mpmath 1.3.0 (independent Cauchy contour-integral implementation)

Properties #

$\lambda_1 = 1 + \gamma/2 - \tfrac12\ln(4\pi)$, exactly

live ↗

In$\mathrm{chop}(N(\mathrm{KeiperLiLambda}(1))-N((1+γ/2)-\ln(4π)/2))$Out$0.0$

Possible issues #

Declines past n = 20 rather than returning a value that has quietly lost precision

live ↗

In$\mathrm{KeiperLiLambda}(21)$Out$\mathrm{KeiperLiLambda}(21)$

About

Examples	24
Crosswalk	FungrimFungrim entry ×3

See also: Zeta