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 = 1/2 axis.

Definition

The Riemann function is given by
where ζ is the Riemann zeta function. Consider the sequence
Li's criterion is then the statement that
The numbers 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
and Im
The positivity of has been verified up to by direct computation.

Proof

Note that.
Then, starting with an entire function, let.
vanishes when. Hence, is holomorphic on the unit disk iff.
Write the Taylor series. Since
we have
so that
Finally, if each zero comes paired with its complex conjugate, then we may combine terms to get
The condition then becomes equivalent to. The right-hand side of is obviously nonnegative when both and . Conversely, ordering the by, we see that the largest term dominates the sum as, and hence becomes negative sometimes.

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