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A059750 Decimal expansion of zeta(1/2) (negated). 20
1, 4, 6, 0, 3, 5, 4, 5, 0, 8, 8, 0, 9, 5, 8, 6, 8, 1, 2, 8, 8, 9, 4, 9, 9, 1, 5, 2, 5, 1, 5, 2, 9, 8, 0, 1, 2, 4, 6, 7, 2, 2, 9, 3, 3, 1, 0, 1, 2, 5, 8, 1, 4, 9, 0, 5, 4, 2, 8, 8, 6, 0, 8, 7, 8, 2, 5, 5, 3, 0, 5, 2, 9, 4, 7, 4, 5, 0, 0, 6, 2, 5, 2, 7, 6, 4, 1, 9, 3, 7, 5, 4, 6, 3, 3, 5, 6, 8, 1 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Zeta(1/2) can be calculated as a limit similar to the limit for the Euler-Mascheroni constant or Euler gamma. - Mats Granvik Nov 14 2012

The WolframAlpha link gives 3 series and 3 integrals for zeta(1/2). To extend the sequence, click "More digits" repeatedly. - Jonathan Sondow, Jun 20 2013

LINKS

Harry J. Smith, Table of n, a(n) for n = 1..5000

J. Sondow and E. W. Weisstein, MathWorld: Riemann Zeta Function

WolframAlpha, zeta(1/2)

FORMULA

Zeta(1/2) = lim_{k->inf} ( Sum_{n=1..k} 1/n^(1/2) - 2*k^(1/2) ) (according to Mathematica 8). - Mats Granvik Nov 14 2012

From Magri Zino, Jan 05 2014 - personal communication: (Start)

The previous result is the case q=2 of the following generalization:

Zeta(1/q) = lim_{k->inf} (Sum_{n=1..k} 1/n^(1/q) - (q/(q-1))*k^((q-1)/q)), with q>1. Example: for q=3/2, Zeta(2/3) = lim_{k->inf} (Sum_{n=1..k} 1/n^(2/3) - 3*k^(1/3)) = -2.447580736233658231... (End)

EXAMPLE

-1.4603545088095868128894991525152980124672293310125814905428860878...

MAPLE

Digits := 120; evalf(Zeta(1/2));

MATHEMATICA

RealDigits[ Zeta[1/2], 10, 111][[1]] (* Robert G. Wilson v, Oct 11 2005 *)

RealDigits[N[Limit[Sum[1/Sqrt[n], {n, 1, k}] - 2*Sqrt[k], k -> Infinity], 90]][[1]] (* Mats Granvik Nov 14 2012 *)

PROG

(PARI) { default(realprecision, 5080); x=-zeta(1/2); for (n=1, 5000, d=floor(x); x=(x-d)*10; write("b059750.txt", n, " ", d)); } \\ Harry J. Smith, Jun 29 2009

CROSSREFS

Cf. A161688 (continued fraction).

Sequence in context: A204017 A021960 A096256 * A243983 A117036 A016723

Adjacent sequences:  A059747 A059748 A059749 * A059751 A059752 A059753

KEYWORD

nonn,cons

AUTHOR

Peter Walker (peterw(AT)aus.ac.ae), Feb 11 2001

EXTENSIONS

Sign of the constant reversed by R. J. Mathar, Feb 05 2009

STATUS

approved

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Last modified December 9 21:41 EST 2016. Contains 278987 sequences.