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A059750
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Decimal expansion of zeta(1/2) (negated).
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25
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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
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OFFSET
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1,2
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COMMENTS
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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
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LINKS
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Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 99.
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FORMULA
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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)
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EXAMPLE
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-1.4603545088095868128894991525152980124672293310125814905428860878...
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MAPLE
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Digits := 120; evalf(Zeta(1/2));
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MATHEMATICA
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RealDigits[N[Limit[Sum[1/Sqrt[n], {n, 1, k}] - 2*Sqrt[k], k -> Infinity], 90]][[1]] (* Mats Granvik Nov 14 2012 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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AUTHOR
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Peter Walker (peterw(AT)aus.ac.ae), Feb 11 2001
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EXTENSIONS
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STATUS
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approved
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