Li's criterion

In number theory, Li's criterion is a particular statement about the positivity of a certain sequence that is equivalent to the Riemann hypothesis. The criterion is named after Xian-Jin Li, who presented it in 1997. In 1999, Enrico Bombieri and Jeffrey C. Lagarias provided a generalization, showing that Li's positivity condition applies to any collection of points that lie on the Re(s) = 1/2 axis.

The Riemann ξ function is given by

where ζ is the Riemann zeta function. Consider the sequence

Li's criterion is then the statement that

The numbers ${\displaystyle \lambda _{n}}$ (sometimes defined with a slightly different normalization) are called Keiper-Li coefficients or Li coefficients. They may also be expressed in terms of the non-trivial zeros of the Riemann zeta function:

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

(Re(s) and Im(s) denote the real and imaginary parts of s, respectively.)

The positivity of ${\displaystyle \lambda _{n}}$ has been verified up to ${\displaystyle n=10^{5}}$ by direct computation.

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