given that Sum[ EulerPhi[k]/k^s , {k,Infinity}] equals Zeta[s-1]/Zeta[s], and
that Sum[ DivisorSigma[p,k]/k^s , {k,Infinity}] equals Zeta[s-p]*Zeta[s],
it is evident that
Sum[(Zeta[3] EulerPhi[k] - DivisorSigma[1, k]/Zeta[3])/k^3, {k,Infinity}] =0
since it boils down to 'Zeta[2]-Zeta[2]'.
Being well aware of the 'indefiniteness \infinitness' of Zeta[1],
I was amused by the behaviour of 'Zeta[1]-Zeta[1]' as in:
Sum[(Zeta[2] EulerPhi[k] - DivisorSigma[1, k]/Zeta[2])/k^2, {k,Infinity}].
Its numerical values dance 'round zero.
I naively expected it to sum to zero anyhow, but it refuses to oblige.
It gets stuck at about 1.1399.. , not a familiar value over at my place.
W.
