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
Examples
0.368790479...
Links
- 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.
Crossrefs
Programs
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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 *)
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