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

Author

Jonathan Vos Post, Feb 14 2007

Keywords

• cons
• nonn

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

Examples

			-0.65699901371692786827912005688957578075547419154089...
		

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.

Crossrefs

Cf. A143301.

Programs

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 *)

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

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

Extensions

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