|
%I #68 Oct 25 2021 16:10:29
%S 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,
%T 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,
%U 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
%N Decimal expansion of zeta(1/2) (negated).
%C 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
%C The WolframAlpha link gives 3 series and 3 integrals for zeta(1/2). - _Jonathan Sondow_, Jun 20 2013
%H Harry J. Smith, <a href="/A059750/b059750.txt">Table of n, a(n) for n = 1..5000</a>
%H B. K. Choudhury, <a href="https://doi.org/10.1098/rspa.1995.0096">The Riemann zeta-function and its derivatives</a>, Proc. R. Soc. Lond A 445 (1995) 477, Table 3.
%H Steven R. Finch, <a href="https://doi.org/10.1017/9781316997741">Mathematical Constants II</a>, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 99.
%H Fredrik Johansson, <a href="https://fredrikj.net/math/zeta_1_2_1m.txt">zeta(1/2) to 1 million digits</a>.
%H Fredrik Johansson, <a href="https://arxiv.org/abs/2110.10583">Rapid computation of special values of Dirichlet L-functions</a>, arxiv:2110.10583 [math.NA], 2021.
%H Hisashi Kobayashi, <a href="https://arxiv.org/abs/1603.02954">Some results on the xi(s) and Xi(t) functions associated with Riemann's zeta(s) function</a>, arXiv preprint arXiv:1603.02954 [math.NT], 2016.
%H Lutz Mattner, Irina Shevtsova, <a href="https://arxiv.org/abs/1710.08503">An optimal Berry-Esseen type theorem for integrals of smooth functions</a>, arXiv:1710.08503 [math.PR], 2017.
%H J. Sondow and E. W. Weisstein, <a href="http://mathworld.wolfram.com/RiemannZetaFunction.html">MathWorld: Riemann Zeta Function</a>
%H WolframAlpha, <a href="http://www.wolframalpha.com/input/?i=zeta%281%2F2%29">zeta(1/2)</a>
%F 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
%F From _Magri Zino_, Jan 05 2014 - personal communication: (Start)
%F The previous result is the case q=2 of the following generalization:
%F 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)
%F Equals -A014176*A113024. - _Peter Luschny_, Oct 25 2021
%e -1.4603545088095868128894991525152980124672293310125814905428860878...
%p Digits := 120; evalf(Zeta(1/2));
%t RealDigits[ Zeta[1/2], 10, 111][[1]] (* _Robert G. Wilson v_, Oct 11 2005 *)
%t RealDigits[N[Limit[Sum[1/Sqrt[n], {n, 1, k}] - 2*Sqrt[k], k -> Infinity], 90]][[1]] (* _Mats Granvik_ Nov 14 2012 *)
%o (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
%Y Cf. A161688 (continued fraction), A078434, A014176, A113024.
%K nonn,cons
%O 1,2
%A Peter Walker (peterw(AT)aus.ac.ae), Feb 11 2001
%E Sign of the constant reversed by _R. J. Mathar_, Feb 05 2009
|
