f(x) divided by 8 gives asymptotic sqrt(f(x))

    (*start*)
    (*Mathematica*)
    Clear[a, f, p];
    nn = 2000;
    p = 1;
    f[n_] := n*Log[n]^p;
    a[n_] := DivisorSum[n, MoebiusMu[#] # &];
    Monitor[TableForm[
       A = Accumulate[
         Table[Table[If[n >= k, a[GCD[n, k]], 0], {k, 1, nn}], {n, 1,
           nn}]]];, n]
    TableForm[B = -Abs[A]];
    Clear[A];
    B[[All, 1]] = N[Table[f[n]/8, {n, 1, nn}]];
    TableForm[B];
    TableForm[B1 = Sign[Transpose[Accumulate[Transpose[B]]]]];
    Clear[B];
    Quiet[Show[
      ListLinePlot[
       v = ReplaceAll[
         Flatten[Table[First[Position[B1[[n]], -1]], {n, 1, nn}]],
         First[{}] -> 1], PlotStyle -> Blue],
      Plot[Sqrt[f[n]], {n, 1, nn}, PlotStyle -> {Red, Thick}],
      ImageSize -> Large]]
    ListLinePlot[v/Table[Sqrt[f[n]], {n, 1, nn}]]
    (*end*)