A126689 Decimal expansion of negative of Granville-Soundararajan constant.

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

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

Named after the British mathematician Andrew James Granville (b. 1962) and the Indian-American mathematician Kannan Soundararajan (b. 1973). - Amiram Eldar, Jun 23 2021

References

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, section 2.33, p. 206.

Links

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

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.

Andrew Granville and Kannan Soundararajan, The spectrum of multiplicative functions, Annals of Mathematics, Vol. 153, No. 2 (2001), pp. 407-470, alternative link.

Formula

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

Equals 1 - 2 * A143301. - Amiram Eldar, Aug 25 2020

Example

-0.65699901371692786827912005688957578075547419154089...

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-François 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, 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

Crossrefs

Cf. A143301.

Sequence in context: A380141 A019621 A335263 * 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