# Keiper-Li coefficients

## Definitions

Symbol: KeiperLiLambda $\lambda_{n}$ Keiper-Li coefficient

## Representations

$\lambda_{n} = \frac{1}{n !} \left[ \frac{d^{n}}{{d s}^{n}} \log\!\left(2 \xi\!\left(\frac{s}{s - 1}\right)\right) \right]_{s = 0}$
$\lambda_{n} = \frac{1}{n} \sum_{\textstyle{k \in \mathbb{Z} \atop k \ne 0}} \left(1 - {\left(\frac{\rho_{k}}{\rho_{k} - 1}\right)}^{n}\right)$

## Specific values

$\lambda_{0} = 0$
$\lambda_{1} = \frac{\gamma}{2} + 1 - \frac{\log\!\left(4 \pi\right)}{2}$
Table of $\lambda_{n}$ to 50 digits for $0 \le n \le 30$
Table of $\lambda_{{10}^{n}}$ to 50 digits for $0 \le n \le 5$

## Asymptotics

$\left(\operatorname{RH}\right) \;\implies\; \left(\lambda_{n} \sim \frac{\log(n)}{2} - \frac{\log\!\left(2 \pi\right) + 1 - \gamma}{2}, \; n \to \infty\right)$

## Riemann hypothesis (Li's criterion)

$\left(\operatorname{RH}\right) \iff \left(\lambda_{n} > 0 \;\text{ for all } n \in \mathbb{Z}_{\ge 1}\right)$
$\left(\operatorname{RH}\right) \iff \left(\sum_{n=1}^{\infty} {\left|\lambda_{n} - a(n)\right|}^{2} < \infty\; \text{ where } a(n) = \frac{\log(n)}{2} - \frac{\log\!\left(2 \pi\right) + 1 - \gamma}{2}\right)$
$\left(\operatorname{Re}\!\left(\rho_{n}\right) = \frac{1}{2} \;\text{ for all } n \in \mathbb{Z}_{\ge 1} \text{ with } \operatorname{Im}\!\left(\rho_{n}\right) < T\right) \;\implies\; \left(\lambda_{n} \ge 0 \;\text{ for all } n \in \mathbb{Z}_{\ge 0} \text{ with } n \le {T}^{2}\right)$

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2021-03-15 19:12:00.328586 UTC