A134472 -id:A134472 - cp's OEIS frontend

A134469 Decimal expansion of -zeta(1/2)/sqrt(2*Pi).

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, 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, 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

Offset: 0

Hans J. H. Tuenter, Oct 27 2007

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.

			0.58259715793901067020517716418763115472909387019865...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 5.9, p. 326.

G. C. Greubel, Table of n, a(n) for n = 0..10000
Joseph T. Chang and Yuval Peres, Ladder heights, Gaussian random walks and the Riemann zeta function, Annals of Probability, 25(2) (1997) 787-802.
Alain Comtet and Satya N. Majumdar, Precise Asymptotics for a Random Walker’s Maximum, J. Stat. Mech. Theor. Exp. 06 (2005) P06013, arXiv:cond-mat/0506195 [cond-mat.stat-mech], 2005.
Hans J. H. Tuenter, Overshoot in the Case of Normal Variables: Chernoff's Integral, Latta's Observation and Wijsman's Sum, Sequential Analysis, 26(4) (2007) 481-488.
Robert A. Wijsman, Overshoot in the Case of Normal Variables, Sequential Analysis, 23(2):275-284, 2004.

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

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

RealDigits[-Zeta[1/2]/Sqrt[2*Pi], 10, 100][[1]] (* G. C. Greubel, Mar 27 2018 *)

-zeta(1/2)/sqrt(2*Pi) \\ Charles R Greathouse IV, Mar 10 2016

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

More decimals from Vaclav Kotesovec, Mar 21 2016

A134470 Continued fraction expansion of -zeta(1/2)/sqrt(2*Pi).

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 8, 1, 5, 1, 1, 1, 12, 5, 1, 1, 5, 1, 12, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 3, 2, 2, 2, 1, 11, 1, 6, 1, 3, 2, 1, 1, 1, 1, 1, 2, 6, 7, 1, 4, 2, 1, 1, 1, 13, 1, 1, 1, 2, 4, 2, 11, 1, 2, 5, 1, 8, 1, 78, 10, 1, 64, 1, 29, 1, 3, 1, 1, 1, 2, 1, 12, 1, 2, 1, 4, 1, 2, 1, 2, 32, 1, 92, 1, 14, 1, 10, 12, 2, 3, 16, 2, 1, 1, 1, 1, 8, 3, 15, 1, 2, 2, 1, 4, 4, 2, 8, 1, 1557, 3, 1, 69, 1, 5, 3, 11, 1, 1

Offset: 0

Hans J. H. Tuenter, Oct 27 2007

G. C. Greubel, Table of n, a(n) for n = 0..10000
Hans J. H. Tuenter, Overshoot in the Case of Normal Variables: Chernoff's Integral, Latta's Observation and Wijsman's Sum, Sequential Analysis, 26(4) (2007) 481-488.

Cf. A134469 (Decimal expansion), A134471 (Numerators of continued fraction convergents), A134472 (Denominators of continued fraction convergents).

Digits:=100; cfrac(-Zeta(1/2)/sqrt(2*Pi),30,'quotients');

ContinuedFraction[ -Zeta[1/2]/Sqrt[2 \[Pi]], 100] (* J. Mulder (jasper.mulder(AT)planet.nl), Jan 25 2010 *)

default(realprecision,1000);
c=-zeta(1/2)/sqrt(2*Pi); /* == 0.582597157... (A134469) */
contfrac(c) /* gives 967 terms */

More terms from J. Mulder (jasper.mulder(AT)planet.nl), Jan 25 2010

A134471 Numerators of the convergents of the continued fraction expansion of -zeta(1/2)/sqrt(2*Pi).

Original entry on oeis.org

0, 1, 1, 3, 4, 7, 60, 67, 395, 462, 857, 1319, 16685, 84744, 101429, 186173, 1032294, 1218467, 15653898, 16872365, 32526263, 49398628, 81924891, 213248410, 295173301, 508421711, 803595012, 1312016723, 3427628458, 11594902097, 26617432652, 64829767401

Offset: 1

Hans J. H. Tuenter, Oct 27 2007

G. C. Greubel, Table of n, a(n) for n = 1..1000
Hans J. H. Tuenter, Overshoot in the Case of Normal Variables: Chernoff's Integral, Latta's Observation and Wijsman's Sum, Sequential Analysis, 26(4) (2007) 481-488.

Cf. A134469 (Decimal expansion), A134470 (Continued fraction expansion), A134472 (Denominators of continued fraction convergents).

Numerator[Convergents[-Zeta[1/2]/Sqrt[2Pi],30]] (* Harvey P. Dale, Sep 07 2015 *)

More terms from Harvey P. Dale, Sep 07 2015

cp's OEIS Frontend

A134469 Decimal expansion of -zeta(1/2)/sqrt(2*Pi).

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A134470 Continued fraction expansion of -zeta(1/2)/sqrt(2*Pi).

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A134471 Numerators of the convergents of the continued fraction expansion of -zeta(1/2)/sqrt(2*Pi).

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