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

Author

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

Keywords

• nonn
• cons

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

Examples

			-1.4603545088095868128894991525152980124672293310125814905428860878...
		

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)

Crossrefs

Cf. A161688 (continued fraction), A078434, A014176, A113024.

Programs

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 *)

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

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

Extensions

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