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A104539 Decimal expansion of lambda(2) in Li's criterion. 19
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 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET
0,2
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.
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
KEYWORD
nonn,cons
AUTHOR
Eric W. Weisstein, Mar 13 2005
STATUS
approved

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Last modified July 25 05:51 EDT 2024. Contains 374586 sequences. (Running on oeis4.)