Li's Criterion - Definition

Definition

The Riemann ξ function is given by

where ζ is the Riemann zeta function. Consider the sequence

$\lambda_n = \frac{1}{(n-1)!} \left. \frac{d^n}{ds^n} \left \right|_{s=1}.$

Li's criterion is then the statement that

the Riemann hypothesis is completely equivalent to the statement that for every positive integer n.

The numbers may also be expressed in terms of the non-trivial zeros of the Riemann zeta function:

$\lambda_n=\sum_{\rho} \left[1- \left(1-\frac{1}{\rho}\right)^n\right]$

where the sum extends over ρ, the non-trivial zeros of the zeta function. This conditionally convergent sum should be understood in the sense that is usually used in number theory, namely, that

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