# 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.

## Definition

The Riemann ξ function is given by

${\displaystyle \xi (s)={\frac {1}{2}}s(s-1)\pi ^{-s/2}\Gamma \left({\frac {s}{2}}\right)\zeta (s)}$

where ζ is the Riemann zeta function. Consider the sequence

${\displaystyle \lambda _{n}={\frac {1}{(n-1)!}}\left.{\frac {d^{n}}{ds^{n}}}\left[s^{n-1}\log \xi (s)\right]\right|_{s=1}.}$

Li's criterion is then the statement that

the Riemann hypothesis is equivalent to the statement that ${\displaystyle \lambda _{n}>0}$ for every positive integer ${\displaystyle n}$.

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:

${\displaystyle \lambda _{n}=\sum _{\rho }\left[1-\left(1-{\frac {1}{\rho }}\right)^{n}\right]}$

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

${\displaystyle \sum _{\rho }=\lim _{N\to \infty }\sum _{|\operatorname {Im} (\rho )|\leq N}.}$

(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.

## A generalization

Bombieri and Lagarias demonstrate that a similar criterion holds for any collection of complex numbers, and is thus not restricted to the Riemann hypothesis. More precisely, let R = {ρ} be any collection of complex numbers ρ, not containing ρ = 1, which satisfies

${\displaystyle \sum _{\rho }{\frac {1+\left|\operatorname {Re} (\rho )\right|}{(1+|\rho |)^{2}}}<\infty .}$

Then one may make several equivalent statements about such a set. One such statement is the following:

One has ${\displaystyle \operatorname {Re} (\rho )\leq 1/2}$ for every ρ if and only if
${\displaystyle \sum _{\rho }\operatorname {Re} \left[1-\left(1-{\frac {1}{\rho }}\right)^{-n}\right]\geq 0}$
for all positive integers n.

One may make a more interesting statement, if the set R obeys a certain functional equation under the replacement s ↦ 1 − s. Namely, if, whenever ρ is in R, then both the complex conjugate ${\displaystyle {\overline {\rho }}}$ and ${\displaystyle 1-\rho }$ are in R, then Li's criterion can be stated as:

One has Re(ρ) = 1/2 for every ρ if and only if
${\displaystyle \sum _{\rho }\left[1-\left(1-{\frac {1}{\rho }}\right)^{n}\right]\geq 0}$
for all positive integers n.

Bombieri and Lagarias also show that Li's criterion follows from Weil's criterion for the Riemann hypothesis.