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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 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, 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->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][] (* Robert G. Wilson v, Oct 11 2005 *) RealDigits[N[Limit[Sum[1/Sqrt[n], {n, 1, k}] - 2*Sqrt[k], k -> Infinity], 90]][] (* 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: A021960 A096256 A319091 * 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 October 14 11:36 EDT 2019. Contains 327996 sequences. (Running on oeis4.)