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A126689 Decimal expansion of negative of Granville-Soundararajan constant. 2
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 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

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

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..2000

Antal Balog, Andrew Granville and K. Soundararajan, Multiplicative functions in arithmetic progressions, arXiv:math.NT/0702389 [math.NT], Feb 13 2007, p. 7.

Andrew Granville and K. Soundararajan, The spectrum of multiplicative functions, Ann. of Math., Vol. 153 (2001), pp. 407-470.

FORMULA

Equals 1-2*log[1+sqrt e]+4*Integral( [log t]/(1+t),t=1..sqrt e) = 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

EXAMPLE

-0.656999...

MAPLE

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

MATHEMATICA

RealDigits[ N[ 4*PolyLog[2, -Sqrt[E]] + Pi^2/3 + 1, 105]][[1]] (* Jean-Fran├žois Alcover, Nov 08 2012, after R. J. Mathar *)

PROG

(PARI) 1-2*log(1+exp(1/2))+4*intnum(t=1, exp(1/2), log(t)/(t+1)) \\ Charles R Greathouse IV, Apr 29 2013

(Python)

from mpmath import *

mp.dps=106

print map(int, list(str(4*polylog(2, -sqrt(e)) + pi**2/3 + 1)[3:-1])) # Indranil Ghosh, Jul 03 2017

CROSSREFS

Sequence in context: A023408 A133616 A019621 * A243093 A101634 A071176

Adjacent sequences:  A126686 A126687 A126688 * A126690 A126691 A126692

KEYWORD

cons,nonn

AUTHOR

Jonathan Vos Post, Feb 14 2007

EXTENSIONS

More terms from R. J. Mathar, Feb 16 2007, Sep 20 2009

STATUS

approved

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Last modified February 16 09:23 EST 2019. Contains 320161 sequences. (Running on oeis4.)