A306341 Decimal expansion of lambda(8) in Li's criterion.

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

Offset

1,2

Links

Table of n, a(n) for n=1..110.

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

1.465755677147060632...

Mathematica

RealDigits[With[{e = EulerGamma, g = StieltjesGamma}, 1 + 4 e - 28 e^2 + 56 e^3 - 70 e^4 + 56 e^5 - 28 e^6 + 8 e^7 - e^8 + 7 Pi^2/2 + 35 Pi^4/48 + 7 Pi^6/240 + 17 Pi^8/161280 - 56 g[1] + 168 e g[1] - 280 e^2 g[1] + 280 e^3 g[1] - 168 e^4 g[1] + 56 e^5 g[1] - 8 e^6 g[1] - 140 g[1]^2 + 280 e g[1]^2 - 252 e^2 g[1]^2 + 112 e^3 g[1]^2 - 20 e^4 g[1]^2 - 56 g[1]^3 + 56 e g[1]^3 - 16 e^2 g[1]^3 - 2 g[1]^4 + 84 g[2] - 140 e g[2] + 140 e^2 g[2] - 84 e^3 g[2] + 28 e^4 g[2] - 4 e^5 g[2] + 140 g[1] g[2] - 168 e g[1] g[2] + 84 e^2 g[1] g[2] - 16 e^3 g[1] g[2] + 28 g[1]^2 g[2] - 12 e g[1]^2 g[2] - 21 g[2]^2 + 14 e g[2]^2 - 3 e^2 g[2]^2 - 2 g[1] g[2]^2 - 140 g[3]/3 + 140/3 e g[3] - 28 e^2 g[3] + 28/3 e^3 g[3] - 4/3 e^4 g[3] - 28 g[1] g[3] + 56/3 e g[1] g[3] - 4 e^2 g[1] g[3] - 4/3 g[1]^2 g[3] + 14/3 g[2] g[3] - 4/3 e g[2] g[3] - g[3]^2/9 + 35 g[4]/3 - 7 e g[4] + 7/3 e^2 g[4] - 1/3 e^3 g[4] + 7/3 g[1] g[4] - 2/3 e g[1] g[4] - 1/6 g[2] g[4] - 7 g[5]/5 + 7/15 e g[5] - 1/15 e^2 g[5] - 1/15 g[1] g[5] + 7 g[6]/90 - 1/90 e g[6] - g[7]/630 - 4 Log[4 Pi] - 49 Zeta[3] - 217 Zeta[5]/4 - 127 Zeta[7]/16], 10, 110][[1]]

Crossrefs

Cf. A074760 (lambda_1), A104539 (lambda_2), A104540 (lambda_3), A104541 (lambda_4).

Cf. A104542 (lambda_5), A306339 (lambda_6), A306340 (lambda_7).

Sequence in context: A366872 A343470 A199289 * A306654 A368358 A114602

Adjacent sequences: A306338 A306339 A306340 * A306342 A306343 A306344

Keyword

nonn,cons

Author

Eric W. Weisstein, Feb 08 2019

Status

approved