A104541 Decimal expansion of lambda(4) in Li's criterion.

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

Offset

0,1

Links

Table of n, a(n) for n=0..101.

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.

Formula

3*Pi^2/4 + Pi^4/96 - 2*log(4) - 2*log(Pi) + 2*gamma - 6*gamma^2 + 4*gamma^3 - gamma^4 - 12*gamma(1) + 12*gamma*gamma(1) - 4*gamma^2*gamma(1) - 2*gamma(1)^2 + 6*gamma(2) - 2*gamma*gamma(2) - 2*gamma(3)/3 - 7*zeta(3)/2 + 1. - Jean-François Alcover, Jul 02 2014

Example

0.368790479...

Mathematica

lambda[n_] := Limit[D[s^(n - 1) Log[RiemannXi[s]], {s, n}], s -> 1]/(n - 1)!; RealDigits[N[lambda[4], 110]][[1]][[1 ;; 102]] (* Jean-François Alcover, Oct 31 2012, after Eric W. Weisstein, updated May 18 2016 *)
RealDigits[With[{e = EulerGamma, g = StieltjesGamma}, 1 + 2 e - 6 e^2 + 4 e^3 - e^4 + 3 Pi^2/4 + Pi^4/96 - 12 g[1] + 12 e g[1] - 4 e^2 g[1] - 2 g[1]^2 + 6 g[2] - 2 e g[2] - 2 g[3]/3 - 2 Log[4 Pi] - 7 Zeta[3]/2], 10, 110][[1]] (* Eric W. Weisstein, Feb 08 2019 *)

Crossrefs

Cf. A074760 (lambda_1), A104539 (lambda_2), A104540 (lambda_3), A104542 (lambda_5).

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

Sequence in context: A248760 A011261 A371749 * A202300 A244467 A200590

Adjacent sequences: A104538 A104539 A104540 * A104542 A104543 A104544

Keyword

nonn,cons,changed

Author

Eric W. Weisstein, Mar 13 2005

Status

approved