Faster higher dimensional sums for the Möbius function

This formulation of the Möbius function:

(*start*)
(*Mathematica*)
nn = 2^7 - 1
Monitor[
 Table[
     +If[n == 2^0, 1, 0]
   - If[n < 2^1, 0, Sum[If[a == n, 1, 0], {a, 2, n/2^0}]]
   + If[n < 2^2, 0,
    Sum[Sum[If[a*b == n, 1, 0], {a, 2, n/2^1}], {b, 2, n/2^1}]]
   - If[n < 2^3, 0,
    Sum[Sum[Sum[If[a*b*c == n, 1, 0], {a, 2, n/2^2}], {b, 2,
       n/2^2}], {c, 2, n/2^2}]]
   + If[n < 2^4, 0,
    Sum[Sum[Sum[
       Sum[If[a*b*c*d == n, 1, 0], {a, 2, n/2^3}], {b, 2, n/2^3}], {c,
        2, n/2^3}], {d, 2, n/2^3}]]
   - If[n < 2^5, 0,
    Sum[Sum[Sum[
       Sum[Sum[If[a*b*c*d*e == n, 1, 0], {a, 2, n/2^4}], {b, 2,
         n/2^4}], {c, 2, n/2^4}], {d, 2, n/2^4}], {e, 2, n/2^4}]]
   + If[n < 2^6, 0,
    Sum[Sum[Sum[
       Sum[Sum[Sum[If[a*b*c*d*e*f == n, 1, 0], {a, 2, n/2^5}], {b, 2,
          n/2^5}], {c, 2, n/2^5}], {d, 2, n/2^5}], {e, 2, n/2^5}], {f,
       2, n/2^5}]]
   - If[n < 2^7, 0,
    Sum[Sum[Sum[
       Sum[Sum[Sum[
          Sum[If[a*b*c*d*e*f*g == n, 1, 0], {a, 2, n/2^6}], {b, 2,
           n/2^6}], {c, 2, n/2^6}], {d, 2, n/2^6}], {e, 2,
        n/2^6}], {f, 2, n/2^6}], {g, 2, n/2^6}]]
   + If[n < 2^8, 0,
    Sum[Sum[Sum[
       Sum[Sum[Sum[
          Sum[Sum[If[a*b*c*d*e*f*g*h == n, 1, 0], {a, 2, n/2^7}], {b,
            2, n/2^7}], {c, 2, n/2^7}], {d, 2, n/2^7}], {e, 2,
         n/2^7}], {f, 2, n/2^7}], {g, 2, n/2^7}], {h, 2, n/2^7}]], {n,
    1, nn}], n]
Monitor[Table[MoebiusMu[n], {n, 1, nn}], n]
%% - %
Count[%, 0]
(*end*)

should be possible to combine with this Hurwitz zeta function:

(*start*)
Clear[a, b, c, sigma, n, s];
nn = 200;
a = 2;
b = 3;
c = 7;
sigma = 1/2;
s = sigma + c*I;
Limit[Sum[1/k^s, {k, 1, a*n}] - Sum[1/k^s, {k, 1, b*n}], n -> 100]
Clear[n];
Show[ListPlot[
  Table[Re[Sum[1/k^s, {k, 1, a*n}] - Sum[1/k^s, {k, 1, b*n}]], {n, 1,
    nn}]],
 ListLinePlot[
  Table[Re[HurwitzZeta[s, b*n + 1] - HurwitzZeta[s, a*n + 1]], {n, 1,
    nn}], PlotStyle -> Red]]
Chop[N[Table[
    Re[Sum[1/k^s, {k, 1, a*n}] - Sum[1/k^s, {k, 1, b*n}]], {n, 1,
     nn}] - Table[
    Re[HurwitzZeta[s, b*n + 1] - HurwitzZeta[s, a*n + 1]], {n, 1,
     nn}]]]
"This transformation given by Mathematica should be possible to use \
in the Möbius function sum:"
Chop[N[Table[Re[Sum[1/k^s, {k, 1, a*n}]], {n, 1, nn}] -
   Table[Re[Zeta[s] - HurwitzZeta[s, a*n + 1]], {n, 1, nn}]]]
(*end*)

The main obstacle though is the missing exact formula for the
Dirichlet Divisor problem, as well as higher dimensional variants
thereof known as the Piltz divisor problem.


(*start*)
(* Feb 9 2022 fungerande Mathematica*)
nn = 2^7 - 1
m1 = Table[+If[n == 2^0,
    1,                                                                \
                                                                      \
          0], {n, 1, nn}]
m2 = Table[-If[n < 2^1, 0,
    Sum[
     If[a      == n,               1, 0], {a, 2,
      n/2^0}]                                     ], {n, 1, nn}]
m3 = Table[+If[n < 2^2, 0,
    Sum[Sum[If[a*b == n,               1, 0], {a, 2, n/2^1}], {b, 2,
      n/2^1}]], {n, 1, nn}]
m4 = Table[-If[n < 2^3, 0,
    Sum[Sum[If[a*b == n, m3[[a]], 0], {a, 2, n/2^1}], {b, 2,
      n/2^1}]], {n, 1, nn}]
m5 = Table[-If[n < 2^4, 0,
    Sum[Sum[If[a*b == n, m4[[a]], 0], {a, 2, n/2^1}], {b, 2,
      n/2^1}]], {n, 1, nn}]
m6 = Table[-If[n < 2^5, 0,
    Sum[Sum[If[a*b == n, m5[[a]], 0], {a, 2, n/2^1}], {b, 2,
      n/2^1}]], {n, 1, nn}]
m7 = Table[-If[n < 2^6, 0,
    Sum[Sum[If[a*b == n, m6[[a]], 0], {a, 2, n/2^1}], {b, 2,
      n/2^1}]], {n, 1, nn}]
m8 = Table[-If[n < 2^7, 0,
    Sum[Sum[If[a*b == n, m7[[a]], 0], {a, 2, n/2^1}], {b, 2,
      n/2^1}]], {n, 1, nn}]
m9 = Table[-If[n < 2^8, 0,
    Sum[Sum[If[a*b == n, m8[[a]], 0], {a, 2, n/2^1}], {b, 2,
      n/2^1}]], {n, 1, nn}]
Accumulate[m1 + m2 + m3 + m4 + m5 + m6 + m7 + m8 + m9]
Monitor[Accumulate[Table[MoebiusMu[n], {n, 1, nn}]], n]
%% - %
Count[%, 0]
M1 = Table[
  Sum[If[Mod[n, k] == 0, m1[[n/k]], 0], {k, 1, nn}], {n, 1, nn}]
M2 = Table[
  Sum[If[Mod[n, k] == 0, m2[[n/k]], 0], {k, 1, nn}], {n, 1, nn}]
M3 = Table[
  Sum[If[Mod[n, k] == 0, m3[[n/k]], 0], {k, 1, nn}], {n, 1, nn}]
M4 = Table[
  Sum[If[Mod[n, k] == 0, m4[[n/k]], 0], {k, 1, nn}], {n, 1, nn}]
M5 = Table[
  Sum[If[Mod[n, k] == 0, m5[[n/k]], 0], {k, 1, nn}], {n, 1, nn}]
M6 = Table[
  Sum[If[Mod[n, k] == 0, m6[[n/k]], 0], {k, 1, nn}], {n, 1, nn}]
M7 = Table[
  Sum[If[Mod[n, k] == 0, m7[[n/k]], 0], {k, 1, nn}], {n, 1, nn}]
M8 = Table[
  Sum[If[Mod[n, k] == 0, m8[[n/k]], 0], {k, 1, nn}], {n, 1, nn}]
M9 = Table[
  Sum[If[Mod[n, k] == 0, m9[[n/k]], 0], {k, 1, nn}], {n, 1, nn}]
M1 + M2 + M3 + M4 + M5 + M6 + M7 + M8 + M9
Accumulate[%]
(*end*)


(*start better*)
(*Feb 9 2022 fungerande Mathematica*)
nn = 2^7 - 1
"1"
m1 = Table[+If[
    n == 2^0,                                                    1,
    0]                                                                \
                          , {n, 1, nn}]
"2"
m2 = Table[-If[n < 2^1, 0,
    Sum[If[      a == n, 1, 0], {a, 2,
      n/2^0}]                    ]                 , {n, 1, nn}]
"3"
m3 = Table[-If[n < 2^2, 0,
    Sum[Sum[If[a*b == n, m2[[a]] , 0], {a, 2, n/2^1}], {b, 2,
      n/2^1}]], {n, 1, nn}]
"4"
m4 = Table[-If[n < 2^3, 0,
    Sum[Sum[If[a*b == n, m3[[a]], 0], {a, 2, n/2^1}], {b, 2,
      n/2^1}]], {n, 1, nn}]
"5"
m5 = Table[-If[n < 2^4, 0,
    Sum[Sum[If[a*b == n, m4[[a]], 0], {a, 2, n/2^1}], {b, 2,
      n/2^1}]], {n, 1, nn}]
"6"
m6 = Table[-If[n < 2^5, 0,
    Sum[Sum[If[a*b == n, m5[[a]], 0], {a, 2, n/2^1}], {b, 2,
      n/2^1}]], {n, 1, nn}]
"7"
m7 = Table[-If[n < 2^6, 0,
    Sum[Sum[If[a*b == n, m6[[a]], 0], {a, 2, n/2^1}], {b, 2,
      n/2^1}]], {n, 1, nn}]
"8"
m8 = Table[-If[n < 2^7, 0,
    Sum[Sum[If[a*b == n, m7[[a]], 0], {a, 2, n/2^1}], {b, 2,
      n/2^1}]], {n, 1, nn}]
"9"
m9 = Table[-If[n < 2^8, 0,
    Sum[Sum[If[a*b == n, m8[[a]], 0], {a, 2, n/2^1}], {b, 2,
      n/2^1}]], {n, 1, nn}]
Accumulate[m1 + m2 + m3 + m4 + m5 + m6 + m7 + m8 + m9]
Monitor[Accumulate[Table[MoebiusMu[n], {n, 1, nn}]], n]
%% - %
(*end better*)