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A126689
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Decimal expansion of negative of Granville-Soundararajan constant.
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3
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6, 5, 6, 9, 9, 9, 0, 1, 3, 7, 1, 6, 9, 2, 7, 8, 6, 8, 2, 7, 9, 1, 2, 0, 0, 5, 6, 8, 8, 9, 5, 7, 5, 7, 8, 0, 7, 5, 5, 4, 7, 4, 1, 9, 1, 5, 4, 0, 8, 9, 8, 3, 1, 6, 5, 7, 1, 5, 7, 7, 8, 1, 6, 3, 5, 2, 6, 0, 2, 7, 8, 8, 8, 1, 1, 3, 8, 2, 8, 4, 4, 0, 2, 4, 0, 5, 7, 6, 0, 3, 8, 2, 6, 3, 9, 8, 3, 6, 5, 3, 8, 7, 1, 5, 2
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OFFSET
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0,1
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COMMENTS
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For any completely multiplicative function f(n) with -1 <= f(n) <= 1, the sum f(1) + f(2) + ... + f(x) is at most (c + o(1))x, where c is this constant. Further, this bound is sharp in that for any c0 > c there are infinitely many f and arbitrarily large x giving a sum less than c0*x. - Charles R Greathouse IV, May 26 2015
Named after the British mathematician Andrew James Granville (b. 1962) and the Indian-American mathematician Kannan Soundararajan (b. 1973). - Amiram Eldar, Jun 23 2021
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REFERENCES
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Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, section 2.33, p. 206.
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LINKS
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Antal Balog, Andrew Granville and Kannan Soundararajan, Multiplicative functions in arithmetic progressions, Annales mathématiques du Québec, Vol. 37, No. 1 (2013), pp. 3-30, see p. 10; arXiv preprint, arXiv:math/0702389 [math.NT], 2007, see p. 7.
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FORMULA
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Equals 1-2*log[1+sqrt e]+4*Integral_{t=1..sqrt e}([log t]/(1+t)) dt = 1-log 4+4*Sum_{s>=1} K(s)/(s*2^s) where K(s)=Sum_{k=0..s} binomial(s,k)*(-1)^k*[exp(k/2)-1]/k. - R. J. Mathar, Feb 16 2007
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EXAMPLE
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-0.65699901371692786827912005688957578075547419154089...
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MAPLE
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Digits := 40 ; K := proc(s) 0.5+add( binomial(s, k)*(-1)^k/k*(exp(0.5*k)-1), k=1..s) ; end: A126689 := proc(smax) 1.0-log(4.0)+add(K(s)*2^(2-s)/s, s=1..smax) ; end: for smax from 0 to 2*Digits do print(A126689(smax)) ; od ; # R. J. Mathar, Feb 16 2007
read("transforms3") ; Digits := 120 : x := 1+Pi^2/3+4*dilog(exp(1/2)+1) ; x := evalf(x) ; CONSTTOLIST(x) ; # R. J. Mathar, Sep 20 2009
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MATHEMATICA
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PROG
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(Python)
from mpmath import mp, polylog, sqrt, e, pi
mp.dps=106
print([int(k) for k in list(str(4*polylog(2, -sqrt(e)) + pi**2/3 + 1)[3:-1])]) # Indranil Ghosh, Jul 03 2017
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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