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 ) {\displaystyle \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
λ n = 1 ( n − 1 ) ! d n d s n [ s n − 1 log ξ ( s ) ] | s = 1 . {\displaystyle \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 equivalent to the statement that λ n > 0 {\displaystyle \lambda _{n}>0} for every positive integer n {\displaystyle n} . The numbers λ n {\displaystyle \lambda _{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 ] {\displaystyle \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
∑ ρ = lim N → ∞ ∑ | Im ( ρ ) | ≤ N . {\displaystyle \sum _{\rho }=\lim _{N\to \infty }\sum _{|\operatorname {Im} (\rho )|\leq N}.}
(Re(s) and Im(s) denote the real and imaginary parts of s, respectively.) The positivity of λ n {\displaystyle \lambda _{n}} has been verified up to n = 10 5 {\displaystyle n=10^{5}} by direct computation.
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