A134469 - OEIS

%I #18 Mar 28 2018 05:29:37

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

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

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

%N Decimal expansion of -zeta(1/2)/sqrt(2*Pi).

%C This number is the limiting expected overshoot over a boundary for the sum of independent and identically distributed normal variables with unit variance, as their positive mean approaches zero. It has applications in sequential analysis.

%H G. C. Greubel, <a href="/A134469/b134469.txt">Table of n, a(n) for n = 0..10000</a>

%H Joseph T. Chang and Yuval Peres, <a href="http://www.jstor.org/stable/2959611">Ladder heights, Gaussian random walks and the Riemann zeta function</a>, Annals of Probability, 25(2) (1997) 787-802.

%H Alain Comtet and Satya N. Majumdar, <a href="http://arxiv.org/abs/cond-mat/0506195">Precise Asymptotics for a Random Walker’s Maximum</a>, J. Stat. Mech. Theor. Exp. 06 (2005) P06013, arXiv:cond-mat/0506195 [cond-mat.stat-mech], 2005.

%H Hans J. H. Tuenter, <a href="http://dx.doi.org/10.1080/07474940701620998">Overshoot in the Case of Normal Variables: Chernoff's Integral, Latta's Observation and Wijsman's Sum</a>, Sequential Analysis, 26(4) (2007) 481-488.

%H Robert A. Wijsman, <a href="http://dx.doi.org/10.1081/SQA-120035933">Overshoot in the Case of Normal Variables</a>, Sequential Analysis, 23(2):275-284, 2004.

%F -zeta(1/2)/sqrt(2*Pi)= A059750/A019727.

%e 0.58259715793901067020517716418763115472909387019865...

%p Digits:=100; evalf(-Zeta(1/2)/sqrt(2*Pi));

%t RealDigits[-Zeta[1/2]/Sqrt[2*Pi], 10, 100][[1]] (* _G. C. Greubel_, Mar 27 2018 *)

%o (PARI) -zeta(1/2)/sqrt(2*Pi) \\ _Charles R Greathouse IV_, Mar 10 2016

%Y Cf. A134470 (continued fraction), A134471 (Numerators of continued fraction convergents), A134472 (Denominators of continued fraction convergents).

%K cons,nonn

%O 0,1

%A _Hans J. H. Tuenter_, Oct 27 2007

%E More decimals from _Vaclav Kotesovec_, Mar 21 2016