Dirichlet divisor problem and sum of roots of unity

(*start*)
nn = 18;
Accumulate[Table[DivisorSigma[0, n], {n, 1, nn - 1}]]
Monitor[Table[-1/2*n + n*HarmonicNumber[n] +
   Sum[(I*Log[-Exp[(2 I n Pi )/k]])/(2*Pi), {k, 1, n}], {n, 2, nn}], n]
FullSimplify[%]
%%% - %
(*end*)

{1, 3, 5, 8, 10, 14, 16, 20, 23, 27, 29, 35, 37, 41, 45, 50, 52}

(*expanded form start*)
TableForm[
 T = Table[
   Table[Chop[N[-1/2 + n/k + I*Log[-E^(2*I*n*Pi/k)]/(2*Pi)]], {k, 1,
     n}], {n, 1, 12}]]
Flatten[Round[T]]
(*end*)

(*start*)
T(n,k) = -1/2 + n/k + I*Log[-E^(2*I*n*Pi/k)]/(2*Pi)
(*end*)


(*more*)
(*expanded form start*)
nn = 12; TableForm[
 T = Table[
   Table[k*FullSimplify[
      0*(-1/2 + n/k) + I*Log[-E^(2*I*n*Pi/k)]/(2*Pi)], {k, 1,
     nn}], {n, 1, nn}]]
TableForm[2*(T[[2 ;; nn, 1 ;; nn]] - Accumulate[Differences[T]])]
TableForm[Accumulate[Differences[T]]]
(*end*)

(*expanded form start*)
nn = 12; TableForm[
 T = Table[
   Table[FullSimplify[(-1/2 + n/k) +
      I*Log[-E^(2*I*n*Pi/k)]/(2*Pi)], {k, 1, nn}], {n, 1, nn}]]
TableForm[
 Table[Table[(-1/2 + n/k) + (k - 2 - 2*Mod[n - 1, k])/(2*k), {k, 1,
    nn}], {n, 1, nn}]]
Clear[n, k, x]
FullSimplify[(-1/2 + n/k) + (k - 2 - 2*x)/(2*k)]
FullSimplify[(-1/2 + (n + 1)/k) + (k - 2 - 2*x)/(2*k)]