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Li's Criterion

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Li's criterion states that the Riemann hypothesis is equivalent to the statement that, for

lambda_n=1/((n-1)!)(d^n)/(ds^n)[s^(n-1)lnxi(s)]|_(s=1),
(1)

where xi(s) is the xi-function, lambda_n>=0 for every positive integer n (Li 1997). Li's constants can be written in alternate form as

lambda_n=((-1)^n)/((n-1)!)(d^n)/(ds^n)[(1-s)^(n-1)lnxi(s)]_(s=0)
(2)

(Coffey 2004).

lambda_n can also be written as a sum of nontrivial zeros rho of zeta(s) as

lambda_n=sum_(rho)[1-(1-1/rho)^n]
(3)

(Li 1997, Coffey 2004), with the sum interpreted as the symmetric limit over |Im[rho]|<=T as T->infty.

A recurrence relation for lambda_n in terms of xi(s) is given by

lambda_(n+1)=lambda_n+1/(n!)[(d^n)/(ds^n)s^n(xi^'(s))/(xi(s))]_(s=1)
(4)

(Coffey 2004).

The first few explicit values of the constants lambda_n are

lambda_1=1+1/2gamma-ln2-1/2lnpi
(5)
lambda_2=1+gamma-gamma^2+1/8pi^2-2gamma_1-2ln2-lnpi
(6)
lambda_3=1+3/2gamma-3gamma^2+gamma^3+3gammagamma_1-6gamma_1+3/2gamma_2+3/8pi^2-7/8zeta(3)-3ln2-3/2lnpi,
(7)

(Coffey 2005, eqn. 8), where gamma is the Euler-Mascheroni constant and gamma_k are Stieltjes constants. lambda_n can be computed efficiently in closed form using recurrence relations due to Coffey (2004), namely

lambda_n=1-1/2n(gamma+lnpi+2ln2)-sum_(m=1)^n(n; m)eta_(m-1)+sum_(m=2)^n(-1)^m(n; m)(1-2^(-m))zeta(m),
(8)

(Coffey 2005, eqn. 10), where, for n>=1,

eta_n=(-1)^(n-1)[(n+1)/(n!)gamma_n+sum_(k=0)^(n-1)((-1)^(k-1))/((n-k-1)!)eta_kgamma_(n-k-1)]
(9)

and eta_0=-gamma.

nlambda_nOEIS
10.0230957...A074760
20.0923457...A104539
30.2076389...A104540
40.3687904...A104541
50.5755427...A104542
71.1244601...A306340
81.4657556...A306341

Edwards (2001, p. 160) gave a numerical value for lambda_1, and numerical values to six digits up to n=25 were tabulated by Coffey (2004).

LiCriterionLambdas

While the values of lambda_n up to n approx 10 are remarkably well fit by a parabola with

lambda_n approx 0.023n^2
(10)

(left figure above), larger terms show clear variation from a parabolic fit (right figure).

See also

Riemann Hypothesis, Riemann Zeta Function Zeros, Xi-Function

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References

Bombieri, E. and Lagarias, J. C. "Complements to Li's Criterion for the Riemann Hypothesis." J. Number Th. 77, 274-287, 1999.Coffey, M. W. "Relations and Positivity Results for Derivatives of the Riemann xi Function." J. Comput. Appl. Math. 166, 525-534, 2004.Coffey, M. W. "Toward Verification of the Riemann Hypothesis: Application of the Li Criterion." Math. Phys. Anal. Geom. 8, 211-255, 2005. https://doi.org/10.1007/s11040-005-7584-9.Edwards, H. M. Riemann's Zeta Function. New York: Dover, 2001.Keiper, J. B. "Power Series Expansions of Riemann's xi Function." Math. Comput. 58, 765-773, 1992.Li, X.-J. "The Positivity of a Sequence of Numbers and the Riemann Hypothesis." J. Number Th. 65, 325-333, 1997.Sloane, N. J. A. Sequences A074760, A104539, A104540, A104541, A104542, A306340, and A306341 in "The On-Line Encyclopedia of Integer Sequences."

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Li's Criterion

Cite this as:

Weisstein, Eric W. "Li's Criterion." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/LisCriterion.html

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