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.
The Riemann ξ function is given by where ζ is the Riemann zeta function.
Consider the sequence Li's criterion is then the statement that The numbers
(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: 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 (Re(s) and Im(s) denote the real and imaginary parts of s, respectively.)
The positivity of
by direct computation.
Then, starting with an entire function
is holomorphic on the unit disk
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
The right-hand side of (1) is obviously nonnegative when both
Conversely, ordering the
) dominates the sum as
"a Li–type criterion for zero–free half-planes of Riemann's zeta function".
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 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
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.
In 2006, P. Freitas proved that all of the zeros of the Riemann zeta function lie inside the region
if and only if the numbers are non-negative for all positive integers
These coefficients are called the
-Li coefficients.
[1] A. Droll[2] generalized the results to the extended Selberg class, A. Bucur, A.-M. Ernvall-Hytönen, A. Odžak and L. Smajlović[3] investigated the behavior of the coefficients for certain functions violating the Riemann hypothesis, and N. Palojärvi[4] proved explicit conditions between finitely many
-Li coefficients and zero-free regions.