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A098468 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).

%I A098468 #33 Feb 16 2025 08:32:54
%S A098468 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 A098468 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 A098468 4,9,8,5,5,9,0,1,8,6,4,9,1,3,1,2,3,9,2,9,8,1,5
%N A098468 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 A098468 B equals EulerGamma - A085609.
%D A098468 Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.7 Euler totient constants, p. 116.
%H A098468 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 A098468 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/TotientSummatoryFunction.html">Totient Summatory Function</a>
%F A098468 Sum_{n=1..N} 1/phi(n) = A*(log(N)+B) + O(log(N)/N). - _Jean-François Alcover_, Apr 28 2018
%e A098468 B = -0.0605742294.../A, where A is A082695.
%t A098468 (* Using S. Finch's notation *)
%t A098468 digits = 102;
%t A098468 A = Zeta[2]*Zeta[3]/Zeta[6];
%t A098468 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 A098468 B = EulerGamma - S;
%t A098468 AB = A*B;
%t A098468 Join[{0}, RealDigits[AB, 10, digits][[1]]] (* _Jean-François Alcover_, Apr 28 2018 *)
%Y A098468 Cf. A082695, A085609.
%K A098468 nonn,cons,hard
%O A098468 0,2
%A A098468 _Eric W. Weisstein_, Sep 09 2004
%E A098468 More digits with the aid of A085609 and A082695 from _R. J. Mathar_, Jul 28 2010
%E A098468 More digits with the aid of A085609 and A082695 from _Vaclav Kotesovec_, Feb 17 2015