Li's Criterion - A Generalization

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

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

One has for every ρ if and only if

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 and are in R, then Li's criterion can be stated as:

One has Re(ρ) = 1/2 for every ρ if and only if

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