Li's criterion

In number theory, Li's criterion is a particular statement about the positivity of a certain sequence that is completely equivalent to the Riemann hypothesis. The criterion is named after Xian-Jin Li, who presented it in 1997. Recently, 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

\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

\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 completely equivalent to the statement that \lambda_n > 0 for every positive integer n.

The numbers \lambda_n 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

\sum_\rho = \lim_{N\to\infty} \sum_{|\Im(\rho)|\le N}.

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

\sum_\rho \frac{1+\left|\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 \Re(\rho) \le 1/2 for every ρ if and only if
\sum_\rho\Re\left[1-\left(1-\frac{1}{\rho}\right)^{-n}\right] \ge 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 \overline{\rho} and 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
\sum_\rho\left[1-\left(1-\frac{1}{\rho}\right)^n \right] \ge 0.

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

References