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

(1)

where is the xi-function, for every positive integer .

The first few values are

			(2)
			(3)

where is the Euler-Mascheroni constant and is a Stieltjes constant. The corresponding numerical values are 0.0230957...(Sloane's A074760; Edwards 2001, p. 160), 0.0923457...(Sloane's A104539), 0.207639...(Sloane's A104540), 0.368790...(Sloane's A104541), and 0.575542...(Sloane's A104542), ....

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 Function." J. Comput. Appl. Math. 166, 525-534, 2004.

Edwards, H. M. Riemann's Zeta Function. New York: Dover, 2001.

Keiper, J. B. "Power Series Expansions of Riemann's 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, and A104542 in "The On-Line Encyclopedia of Integer Sequences."

CITE THIS AS:

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