The Floor function and roots of unity

Clear[n, k, s, nn];
nn = 22;
FullSimplify[
 Table[Sum[-1/2 + (1 + n)/k +
    I/(2*Pi)*Log[-Exp[2*Pi*I*(1 + n)/k]], {k, 1, n}], {n, 1, nn}]]
Accumulate[DivisorSigma[0, Range[nn]]]
Chop[N[%% - %]]

Clear[n, k, s, nn];
nn = 22;
FullSimplify[
 Table[Table[-1/2 + (1 + n)/k +
    I/(2*Pi)*Log[-((-1)^(2*(1 + n)/k))], {k, 1, nn}], {n, 1, nn}]]
Table[Table[Floor[n/k], {k, 1, nn}], {n, 1, nn}]
Chop[N[%% - %]]

https://math.stackexchange.com/a/4445905/8530