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

Read more about this topic: Li's Criterion

Famous quotes containing the word definition:

“One definition of man is “an intelligence served by organs.””
—Ralph Waldo Emerson (1803–1882)

“Was man made stupid to see his own stupidity?
Is God by definition indifferent, beyond us all?
Is the eternal truth man’s fighting soul
Wherein the Beast ravens in its own avidity?”
—Richard Eberhart (b. 1904)

“Scientific method is the way to truth, but it affords, even in
principle, no unique definition of truth. Any so-called pragmatic
definition of truth is doomed to failure equally.”
—Willard Van Orman Quine (b. 1908)