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%I #74 Mar 27 2024 20:11:05
%S 0,2,3,0,9,5,7,0,8,9,6,6,1,2,1,0,3,3,8,1,4,3,1,0,2,4,7,9,0,6,4,9,5,2,
%T 9,1,6,2,1,9,3,2,1,2,7,1,5,2,0,5,0,7,5,9,5,2,5,3,9,2,0,7,2,2,1,2,9,7,
%U 1,3,5,6,4,7,6,7,2,4,5,7,9,9,7,0,7,9,8,5,6,9,5,1,1,7,0,9,8,3,3,3,6,4,3,0
%N Decimal expansion of lambda(1) in Li's criterion.
%C Decimal expansion of -B =(1/2)*sum(r in Z, 1/r/(1-r)) where Z is the set of zeros of the Riemann zeta function which lie in the strip 0 <= Re(z) <= 1.
%C According to Gun, Murty, & Rath (2018), it is not even known whether this constant is rational or not (though see Theorem 3.1), though they show that it is transcendental under Schanuel’s conjecture. - _Charles R Greathouse IV_, Nov 12 2021
%D H. M. Edwards, Riemann's Zeta Function, Dover Publications Inc. 1974, p. 160.
%D S. J. Patterson, "An introduction to the theory of the Riemann Zeta-function", Cambridge Studies in Advanced Mathematics 14, p. 34.
%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 Sanoli Gun, M. Ram Murty, and Purusottam Rath, <a href="https://arxiv.org/abs/1807.11201">Transcendental sums related to the zeros of zeta functions</a>, arXiv:1807.11201 [math.NT], 2018; Mathematika, Vol. 64, no. 3 (2018), pp. 875-897.
%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 J. Sondow and C. Dumitrescu, <a href="http://arxiv.org/abs/1005.1104">A monotonicity property of Riemann's xi function and a reformulation of the Riemann Hypothesis</a>, arXiv:1005.1104 [math.NT], 2010; see p. 3 in the link.
%H J. Sondow and C. Dumitrescu, <a href="https://doi.org/10.1007/s10998-010-1037-3">A monotonicity property of Riemann's xi function and a reformulation of the Riemann Hypothesis</a>, Periodica Math. Hungarica, 60 (2010), 37-40; see p. 39 in the link.
%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>.
%H <a href="/index/Z#zeta_function">Index entries for zeta function</a>.
%F -B = Gamma/2 + 1 - log(4*Pi)/2 = 0.0230957...
%e 0.023095708966121033814310247906495291621932127152050759525392...
%t RealDigits[EulerGamma/2 + 1 - Log[4 Pi]/2, 10, 110][[1]]
%o (PARI) Euler/2+1-log(4*Pi)/2 \\ _Charles R Greathouse IV_, Jan 26 2012
%Y Cf. A002410 (nearest integer to imaginary part of n-th zeta zero), A195423 (twice the constant).
%Y Cf. A104539 (lambda_2), A104540 (lambda_3), A104541 (lambda_4), A104542 (lambda_5).
%Y Cf. A306339 (lambda_6), A306340 (lambda_7), A306341 (lambda_8).
%K cons,nonn
%O 0,2
%A _Benoit Cloitre_, Sep 28 2002
%E Name simplified by _Eric W. Weisstein_, Feb 08 2019
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