Linear programming Square root bounds and Harmonic numbers

(*Numerators of the partial sums of the Möbius inverse of the Harmonic numbers*)
(*start*)
Clear[T, n, k, a];
nn = 7;
a[n_] := If[n < 1, 0, Sum[d MoebiusMu@d, {d, Divisors[n]}]]
TableForm[
 M = Table[
   Table[Sum[If[n >= k, a[GCD[n, k]], 0], {n, 1, m}], {k, 1, nn}], {m,
     1, nn}]]
(*end*)

(*The partial sums of the Möbius inverse of the Harmonic numbers fit within this linear programming problem*)
(*start*)
nn = 42;
TableForm[
  L1 = Table[
    LinearProgramming[
     Table[1/n, {n, 1, k}], {Table[If[n == 1, k, 1], {n, 1, k}]}, {{1,
        0}}, Table[
      If[n == 1, {-1, 1}, {-2 (n - 1), 2 (n - 1)}], {n, 1, k}]], {k,
     1, nn}]];
b = Table[Sum[-Sign[-1 + Sign[L1[[n, k]]]], {k, 1, n}], {n, 1, nn}]
(*end*)

(*Setting the upper bound on the variables to zero gives us a square root bound, but then it is no longer an
arithmetic sequence.*)
(*start*)
nn = 42;
TableForm[
  L2 = Table[
    LinearProgramming[
     Table[1/n, {n, 1, k}], {Table[If[n == 1, k, 1], {n, 1, k}]}, {{1,
        0}}, Table[
      If[n == 1, {-1, 1}, {-2 (n - 1), 0}], {n, 1, k}]], {k, 1, nn}]];
c = Table[1 - Sum[Sign[L2[[n, k]]], {k, 1, n}], {n, 1, nn}]
(*end*)

(*Numerators of the partial sums of the Möbius inverse of the \
Harmonic numbers*)
(*start*)
Clear[T, n, k, a, b];
nn = 8;
a[n_] := If[n < 1, 0, Sum[d MoebiusMu@d, {d, Divisors[n]}]]
TableForm[
  Table[Table[If[n >= k, a[GCD[n, k]], 0], {k, 1, nn}], {n, 1, nn}]];
TableForm[
  M = Table[
    Table[If[n >= k, StringJoin["a", ToString[n], ToString[k]],
      0], {k, 1, nn}], {n, 1, nn}]];
TableForm[
  Table[Table[
    Table[M[[m, k]], {m, n + k - 1, n + 2*(k - 1)}], {n, 1,
     nn - 2*(k - 1)}], {k, 1, nn}]];
TableForm[
  Table[Table[
    Table[StringJoin[
      ToString[Sum[M[[m, k]], {m, n + k - 1, n + 2*(k - 1)}]],
      If[k == 1, "==1", StringJoin["==", ToString[0*2*(k - 1)]]]], {n,
       1, nn - 2*(k - 1)}], {k, 1, nn}], {nn, 1, nn}]];
TableForm[
  B = Table[
    Table[StringJoin[
      ToString[
       Sum[Sum[M[[m, k]], {m, n + k - 1, n + 2*(k - 1)}], {n, 1,
         nn - 2*(k - 1)}]],
      If[k == 1, "==1", StringJoin["<=", ToString[2*(k - 1)]]]], {k,
      1, nn}], {nn, 1, nn}]];
"Some of the constraints"
TableForm[
 B1 = Table[
   Table[StringJoin[
     ToString[
      Sum[Sum[M[[m, k]], {m, n + k - 1, n + 2*(k - 1)}], {n, 1,
        nn - 2*(k - 1)}]], If[k == 1, "==1", "<="],
     ToString[2*(k - 1)]], {k, 1, nn}], {nn, 1, nn}]]
TableForm[
 B1 = Table[
   Table[StringJoin[
     ToString[
      Sum[Sum[M[[m, k]], {m, n + k - 1, n + 2*(k - 1)}], {n, 1,
        nn - 2*(k - 1)}]], If[k == 1, "==1", "=>"],
     ToString[-2*(k - 1)]], {k, 1, nn}], {nn, 1, nn}]]
TableForm[
  B1 = Table[
    Table[StringJoin[
      ToString[
       Sum[Sum[M[[m, k]], {m, n + k - 1, n + 2*(k - 1)}], {n, 1,
         nn - 2*(k - 1)}]]], {k, 1, nn}], {nn, 1, nn}]];
TableForm[
 B2 = Table[
   StringJoin[
    ToString[Sum[StringJoin[ToString[B1[[n, k]]]], {k, 1, n}]],
    "==1"], {n, 1, nn}]]
"Maximize this:"
TableForm[
 B3 = Table[
   StringJoin[
    ToString[
     Sum[StringJoin["(", ToString[B1[[n, k]]], ")/", ToString[k]], {k,
        1, n}]]], {n, 1, nn}]]
(*end*)

(*start*)
"For larger epsilons we get non zero differences"
epsilon = 1/100;
Table[Floor[x^(1/2 + epsilon) + 1/2] - Floor[x^(1/2) + 1/2], {x, 0,
  400}]

"For smaller epsilons we get zero"
epsilon = 1/100000000;
Table[Floor[x^(1/2 + epsilon) + 1/2] - Floor[x^(1/2) + 1/2], {x, 0,
  400}]
(*end*)