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).

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

Offset: 0

Author

Eric W. Weisstein, Sep 09 2004

Keywords

• nonn
• cons
• hard

Comments

B equals EulerGamma - A085609.

Examples

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

References

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

Links

• Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 52 (Z1*(gamma-Z2)).
• Eric Weisstein's World of Mathematics, Totient Summatory Function

Crossrefs

Cf. A082695, A085609.

Programs

Mathematica

(* Using S. Finch's notation *)
digits = 102;
A = Zeta[2]*Zeta[3]/Zeta[6];
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]&;
B = EulerGamma - S;
AB = A*B;
Join[{0}, RealDigits[AB, 10, digits][[1]]] (* Jean-François Alcover, Apr 28 2018 *)

Formula

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

Extensions

More digits with the aid of A085609 and A082695 from R. J. Mathar, Jul 28 2010

More digits with the aid of A085609 and A082695 from Vaclav Kotesovec, Feb 17 2015