A104540 - OEIS

%I #29 Nov 18 2019 01:34:02

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

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

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

%N Decimal expansion of lambda(3) in Li's criterion.

%H E. Bombieri and J. C. Lagarias, <a href="https://doi.org/10.1006/jnth.1999.2392">Complements to Li's Criterion for the Riemann Hypothesis</a>, J. Number Th. 77(2) (1999), 274-287.

%H M. W. Coffey, <a href="https://doi.org/10.1016/j.cam.2003.09.003">Relations and positivity results for derivatives of the Riemann xi function</a>, J. Comput. Appl. Math. 166(2) (2004), 525-534.

%H Xian-Jin Li, <a href="https://doi.org/10.1006/jnth.1997.2137">The positivity of a sequence of numbers and the Riemann hypothesis</a>, J. Number Th. 65(2) (1997), 325-333.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LisCriterion.html">Li's Criterion</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RiemannZetaFunctionZeros.html">Riemann Zeta Function Zeros</a>.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Li%27s_criterion">Li's criterion</a>.

%F lambda(3) = 3*Pi^2/8 - 3*log(2) - 3*log(Pi)/2 + 3*gamma/2 - 3*gamma^2 + gamma^3 + 3*gamma*gamma(1) - 6*gamma(1) + 3*gamma(2)/2 - 7*zeta(3)/8 + 1. - _Jean-François Alcover_, Jul 02 2014

%e 0.207638920...

%t lambda[n_] := Limit[D[s^(n - 1) Log[RiemannXi[s]], {s, n}], s -> 1]/(n - 1)!; RealDigits[N[lambda[3], 110]][[1]][[1 ;; 102]] (* _Jean-François Alcover_, Oct 31 2012, after _Eric W. Weisstein_, updated May 18 2016 *)

%t RealDigits[With[{e = EulerGamma, g = StieltjesGamma}, 1 + 3 e/2 - 3 e^2 + e^3 + 3 Pi^2/8 - 6 g[1] + 3 e g[1] + 3 g[2]/2 - Log[8] - 3 Log[Pi]/2 - 7 Zeta[3]/8], 10, 110][[1]] (* _Eric W. Weisstein_, Feb 08 2019 *)

%Y Cf. A074760 (lambda_1), A104539 (lambda_2), A104541 (lambda_4), A104542 (lambda_5).

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

%K nonn,cons

%O 0,1

%A _Eric W. Weisstein_, Mar 13 2005