Li's criterion - Alchetron, The Free Social Encyclopedia

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

ξ ( s ) = 1 2 s ( s − 1 ) π − s / 2 Γ ( s 2 ) ζ ( s )

where ζ is the Riemann zeta function. Consider the sequence

λ n = 1 ( n − 1 ) ! d n d s n [ s n − 1 log ⁡ ξ ( s ) ] | s = 1 .

Li's criterion is then the statement that

the Riemann hypothesis is equivalent to the statement that λ n > 0 for every positive integer n .

The numbers λ 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:

λ n = ∑ ρ [ 1 − ( 1 − 1 ρ ) n ]

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

∑ ρ = lim N → ∞ ∑ | Im ⁡ ( ρ ) | ≤ N .

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

The positivity of λ n has been verified up to 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

∑ ρ 1 + | Re ⁡ ( ρ ) | ( 1 + | ρ | ) 2 < ∞ .

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

One has Re ⁡ ( ρ ) ≤ 1 / 2 for every ρ if and only if ∑ ρ Re ⁡ [ 1 − ( 1 − 1 ρ ) − n ] ≥ 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 ρ ¯ and 1 − ρ are in R, then Li's criterion can be stated as:

One has Re(ρ) = 1/2 for every ρ if and only iffor all positive integers n.

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

References

Li's criterion Wikipedia

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Contents

Definition

A generalization

References