A059750 Decimal expansion of zeta(1/2) (negated).

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

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). - Jonathan Sondow, Jun 20 2013

References

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.6.3, p. 43.

Links

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

B. K. Choudhury, The Riemann zeta-function and its derivatives, Proc. R. Soc. Lond A 445 (1995) 477, Table 3.

Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 99.

Fredrik Johansson, zeta(1/2) to 1 million digits.

Fredrik Johansson, Rapid computation of special values of Dirichlet L-functions, arxiv:2110.10583 [math.NA], 2021.

Hisashi Kobayashi, Some results on the xi(s) and Xi(t) functions associated with Riemann's zeta(s) function, arXiv preprint arXiv:1603.02954 [math.NT], 2016.

Lutz Mattner and Irina Shevtsova, An optimal Berry-Esseen type theorem for integrals of smooth functions, arXiv:1710.08503 [math.PR], 2017.

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

WolframAlpha, zeta(1/2)

Formula

zeta(1/2) = lim_{k->oo} ( 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->oo} (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->oo} (Sum_{n=1..k} 1/n^(2/3) - 3*k^(1/3)) = -2.447580736233658231... (End)

Equals -A014176*A113024. - Peter Luschny, Oct 25 2021

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), A078434, A014176, A113024.

Sequence in context: A096256 A319091 A328227 * A372919 A243983 A117036

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