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)
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
KEYWORD
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