A104539 Decimal expansion of lambda(2) in Li's criterion.

0, 9, 2, 3, 4, 5, 7, 3, 5, 2, 2, 8, 0, 4, 6, 6, 7, 0, 3, 8, 5, 7, 2, 8, 4, 8, 6, 1, 9, 2, 0, 6, 7, 8, 8, 6, 7, 7, 4, 1, 3, 2, 2, 1, 6, 6, 2, 8, 2, 4, 6, 5, 0, 9, 3, 9, 9, 6, 3, 2, 5, 9, 7, 9, 3, 3, 9, 8, 5, 3, 8, 9, 2, 0, 3, 1, 1, 6, 1, 1, 5, 4, 1, 1, 7, 2, 9, 4, 0, 2, 3, 4, 6, 2, 1, 0, 7, 4, 7, 6, 1, 1, 7

Offset

0,2

Links

Table of n, a(n) for n=0..102.

E. Bombieri and J. C. Lagarias, Complements to Li's Criterion for the Riemann Hypothesis, J. Number Th. 77(2) (1999), 274-287.

M. W. Coffey, Relations and positivity results for derivatives of the Riemann xi function, J. Comput. Appl. Math. 166(2) (2004), 525-534.

Xian-Jin Li, The positivity of a sequence of numbers and the Riemann hypothesis, J. Number Th. 65(2) (1997), 325-333.

Eric Weisstein's World of Mathematics, Li's Criterion.

Eric Weisstein's World of Mathematics, Riemann Zeta Function Zeros.

Wikipedia, Li's criterion.

Example

0.0923457352...

Mathematica

lambda[n_] := Limit[D[s^(n - 1)*Log[RiemannXi[s]], {s, n}], s -> 1]/(n - 1)!; Join[{0}, RealDigits[lambda[2], 10, 102] // First]
lambda[2] = 1 + EulerGamma - EulerGamma^2 + Pi^2/8 - Log[4 Pi] - 2*StieltjesGamma[1]; Join[{0}, RealDigits[lambda[2], 10, 102] // First] (* Jean-François Alcover, Oct 31 2012, after Eric W. Weisstein, updated May 18 2016 *)
RealDigits[With[{e = EulerGamma, g = StieltjesGamma}, 1 + e - e^2 + Pi^2/8 - 2 g[1] - Log[4 Pi]], 10, 110, -1][[1]] (* Eric W. Weisstein, Feb 08 2019 *)

Crossrefs

Cf. A074760 (lambda_1), A104540 (lambda_3), A104541 (lambda_4), A104542 (lambda_5).

Cf. A306339 (lambda_6), A306340 (lambda_7), A306341 (lambda_8).

Sequence in context: A291363 A010161 A222226 * A201559 A300015 A246499

Adjacent sequences: A104536 A104537 A104538 * A104540 A104541 A104542

Keyword

nonn,cons

Author

Eric W. Weisstein, Mar 13 2005

Status

approved