OFFSET

0,1

COMMENTS

The squarefree totient analog of the harmonic number F_n is given by F_n = Sum_{k=1..n} mu^2(k)/phi(k) where mu(k) is the Möbius function and phi(k) is Euler's totient function.

Conjectured to be equivalent to Mertens's constant B_3 minus Euler's constant (A138312). B_3 - gamma is given by Sum_{i>=1} log p_i/(p_i*(p_i-1)), where p_i is the i^th prime = Sum_{j>=2} mu(j)*zeta'(j)/zeta(j), mu(j) is the Mobius function, zeta'(j) is the derivative of zeta(j).

LINKS

FORMULA

Limit_{n -> infinity} ((Sum_{k=1..n} mu^2(k)/phi(k)) - H_n), where mu(k) is the Möbius function, phi(k) is Euler's totient function and H_n is the n-th harmonic number.

EXAMPLE

0.755366...

MATHEMATICA

<< NumberTheory`NumberTheoryFunctions` prl = 100000; ts = 0; f = 1; While[f < 100000000000, If[SquareFreeQ[f], ts += N[1/EulerPhi[f], 15]; If[f > prl, Print[{f, ts, hn = N[HarmonicNumber[f], 15], N[ts - hn, 10]}]; prl += 100000]]; f += 1]

CROSSREFS

KEYWORD

AUTHOR

Dick Boland (abstract(AT)imathination.org), Mar 13 2008

STATUS

approved