Li's criterion

Xian-Jin Li [ MR 98d:11101] proved the following assertion: The Riemann Hypothesis is true if and only if $\lambda_n\ge 0$ for each $n=1,2,\dots$ where

$$\lambda_n =\sum_\rho (1-(1-1/rho)^n)$$

Another expression for $\lambda_n$ is given by

$$\lambda_n=\frac{1}{(n-1)!}\frac{d^n}{ds^n}(s^{n-1}\log \xi(s))\vert_{s=1}$$

and

$$\xi(s)=\frac12s(s-1)\Gamma(s/2)\zeta(s)$$

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