%I #32 Sep 24 2021 01:27:00

%S 0,6,0,5,7,4,2,2,9,4,8,6,3,0,5,7,3,2,1,6,0,9,7,4,4,0,1,1,6,6,3,1,3,8,

%T 4,0,3,5,4,9,7,2,2,8,4,0,8,8,2,9,8,9,2,8,1,1,5,1,2,2,4,4,8,5,6,0,9,3,

%U 4,9,8,5,5,9,0,1,8,6,4,9,1,3,1,2,3,9,2,9,8,1,5

%N Decimal expansion of constant A*B in the asymptotic expression of the summatory function Sum_{n=1..N} (1/phi(n)) as A(log(N)+B) + O(log(N)/N).

%C B equals EulerGamma - A085609.

%D Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.7 Euler totient constants, p. 116.

%H Steven R. Finch, <a href="https://doi.org/10.1017/9781316997741">Mathematical Constants II</a>, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 52 (Z1*(gamma-Z2)).

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TotientSummatoryFunction.html">Totient Summatory Function</a>

%F Sum_{n=1..N} 1/phi(n) = A*(log(N)+B) + O(log(N)/N). - _Jean-François Alcover_, Apr 28 2018

%e B = -0.0605742294.../A, where A is A082695.

%t (* Using S. Finch's notation *)

%t digits = 102;

%t A = Zeta[2]*Zeta[3]/Zeta[6];

%t S = Sum[Switch[Mod[k, 6], 0, 1, 1, 0, 2, -1, 3, -1, 4, 0, 5, 1]*PrimeZetaP'[k], {k, 2, 400}] // N[#, digits+40]&;

%t B = EulerGamma - S;

%t AB = A*B;

%t Join[{0}, RealDigits[AB, 10, digits][[1]]] (* _Jean-François Alcover_, Apr 28 2018 *)

%Y Cf. A082695, A085609.

%K nonn,cons,hard

%O 0,2

%A _Eric W. Weisstein_, Sep 09 2004

%E More digits with the aid of A085609 and A082695 from _R. J. Mathar_, Jul 28 2010

%E More digits with the aid of A085609 and A082695 from _Vaclav Kotesovec_, Feb 17 2015