Linear programming and the Riemann hypothesis

(*start*)
(*Mathematica 8.0.1*)
Clear[nn, constant, s, a, d, M, T, n, k, m, sequenceToBeBounded,
  linearProgrammingSolution];
nn = 50;
constant = 1;
s = 1;
a[n_] := If[n < 1, 0, Sum[d MoebiusMu@d, {d, Divisors[n]}]]

Monitor[TableForm[
   M = Table[
     Table[Sum[If[m >= k, a[GCD[m, k]], 0], {m, 1, n}], {k, 1,
       nn}], {n, 1, nn}]];, n]
Monitor[sequenceToBeBounded =
  Table[Sum[M[[n, k]]/k^s, {k, 2, n}], {n, 1,
    nn}](*<--sequence to be bounded*), n]
Table[Sum[
  Sum[If[Mod[m, k] == 0, MoebiusMu[m/k]*HarmonicNumber[k], 0], {k, 1,
     m}] - 1, {m, 1, n}], {n, 1, nn}]
%% - %

"1"
Monitor[TableForm[
  PartialSumsOfMöbiusInverseOfHarmonicNumberLinearProgrammingSolution \
= Table[LinearProgramming[
     Table[1/k^s, {k, 1, n}], {Table[
       If[k == 1, n, 1], {k, 1, n}]}, {{1, 0}},
     Table[If[k == 1, {-1, 1}, {M[[n, k]], M[[n, k]]}], {k, 1,
       n}]], {n, 1, nn}]], n]
Monitor[lowerBound1 =
   Table[Sum[
     PartialSumsOfMöbiusInverseOfHarmonicNumberLinearProgrammingSoluti\
on[[n, k]]/k^s, {k, 2, n}], {n, 1, nn}];, n]

Show[ListPlot[constant*lowerBound1, PlotStyle -> {Thick},
  PlotMarkers -> Automatic],
 ListLinePlot[constant*lowerBound1, PlotStyle -> {Thick}],
 ListLinePlot[sequenceToBeBounded, PlotStyle -> {Red, Thick}],
 ImageSize -> Large]

"2"
constant = 2;
Monitor[TableForm[
  AbsoluteValueNumberTheoreticLinearProgrammingSolution =
   Table[LinearProgramming[
     Table[1/k^s, {k, 1, n}], {Table[
       If[k == 1, n, 1], {k, 1, n}]}, {{1, 0}},
     Table[If[k == 1, {-1, 1}, {-Abs[M[[n, k]]], 0 (k - 1)}], {k, 1,
       n}]], {n, 1, nn}]], n]
Monitor[lowerBound2 =
   Table[Sum[
     AbsoluteValueNumberTheoreticLinearProgrammingSolution[[n, k]]/
      k^s, {k, 2, n}], {n, 1, nn}];, n]

"3"
constant = 2;
Monitor[TableForm[
   OrdinaryLinearProgrammingSolution =
    Table[LinearProgramming[
      Table[1/k^s, {k, 1, n}], {Table[
        If[k == 1, n, 1], {k, 1, n}]}, {{1, 0}},
      Table[If[
        k == 1, {-1, 1}, {-(k - 1) + 0*Abs[M[[n, k]]],
         0 (k - 1)}], {k, 1, n}]], {n, 1, nn}]];, n]
Monitor[lowerBound3 =
   Table[Sum[
     OrdinaryLinearProgrammingSolution[[n, k]]/k^s, {k, 2, n}], {n, 1,
      nn}];, n]

Show[ListLinePlot[constant*lowerBound3, PlotStyle -> {Thick}],
 ListPlot[constant*lowerBound3, PlotStyle -> {Thick},
  PlotMarkers -> Automatic],
 ListPlot[lowerBound1, PlotStyle -> {Thick, Red},
  PlotMarkers -> Automatic],
 ListLinePlot[lowerBound1, PlotStyle -> {Thick, Red}],
 ListPlot[constant*lowerBound2, PlotStyle -> {Thick},
  PlotMarkers -> Automatic],
 ListLinePlot[constant*lowerBound2, PlotStyle -> {Thick},
  PlotMarkers -> Automatic], ImageSize -> Large]
(*end*)


(*****************************************************)
(*****************************************************)
(*****************************************************)

(*start*)(*Mathematica 8.0.1*)
Clear[nn, constant, s, a, d, M, T, n, k, m, sequenceToBeBounded,
  linearProgrammingSolution];
nn = 50;
constant = 1;
s = 1;
a[n_] := If[n < 1, 0, Sum[d MoebiusMu@d, {d, Divisors[n]}]]

Monitor[TableForm[
   M = Table[
     Table[Sum[If[m >= k, a[GCD[m, k]], 0], {m, 1, n}], {k, 1,
       nn}], {n, 1, nn}]];, n]
Monitor[sequenceToBeBounded =
   Table[Sum[M[[n, k]]/k^s, {k, 2, n}], {n, 1,
     nn}];(*<--sequence to be bounded*), n]
Monitor[sequenceToBeBounded2 =
   Table[Sum[(M[[n, k]] - 1)/k^s, {k, 2, n}], {n, 1,
     nn}];(*<--sequence to be bounded*), n]
Table[Sum[
   Sum[If[Mod[m, k] == 0, MoebiusMu[m/k]*HarmonicNumber[k], 0], {k, 1,
       m}] - 1, {m, 1, n}], {n, 1, nn}];
(*%%-%;*)

"1"
Monitor[TableForm[
   PartialSumsOfMöbiusInverseOfHarmonicNumberLinearProgrammingSolution\
 = Table[LinearProgramming[
      Table[1/k^s, {k, 1, n}], {Table[
        If[k == 1, n, 1], {k, 1, n}]}, {{1, 0}},
      Table[If[k == 1, {-1, 1}, {M[[n, k]] - 1, M[[n, k]]}], {k, 1,
        n}]], {n, 1, nn}]];, n]
Monitor[lowerBound1 =
   Table[Sum[
     PartialSumsOfMöbiusInverseOfHarmonicNumberLinearProgrammingSoluti\
on[[n, k]]/k^s, {k, 2, n}], {n, 1, nn}];, n]

Show[ListPlot[constant*lowerBound1, PlotStyle -> {Thick},
  PlotMarkers -> Automatic],
 ListLinePlot[constant*lowerBound1, PlotStyle -> {Thick}],
 ListLinePlot[sequenceToBeBounded, PlotStyle -> {Red, Thick}],
 ImageSize -> Large]

"2"
constant = 1;
Monitor[TableForm[
   AbsoluteValueNumberTheoreticLinearProgrammingSolution =
    Table[LinearProgramming[
      Table[1/k^s, {k, 1, n}], {Table[
        If[k == 1, n, 1], {k, 1, n}]}, {{1, 0}},
      Table[If[k == 1, {-1, 1}, {-Abs[M[[n, k]]] - 1, 0 (k - 1)}], {k,
         1, n}]], {n, 1, nn}]];, n]
Monitor[lowerBound2 =
   Table[Sum[
     AbsoluteValueNumberTheoreticLinearProgrammingSolution[[n, k]]/
      k^s, {k, 2, n}], {n, 1, nn}];, n]
"here here here"
Show[ListLinePlot[lowerBound2, PlotStyle -> Blue],
 ListLinePlot[sequenceToBeBounded2, PlotStyle -> Red]]

"3"
constant = 1;
Monitor[TableForm[
   OrdinaryLinearProgrammingSolution =
    Table[LinearProgramming[
      Table[1/k^s, {k, 1, n}], {Table[
        If[k == 1, n, 1], {k, 1, n}]}, {{1, 0}},
      Table[If[
        k == 1, {-1, 1}, {-k - 1 + 0*Abs[M[[n, k]]], 0 (k - 1)}], {k,
        1, n}]], {n, 1, nn}]];, n]
Monitor[lowerBound3 =
   Table[Sum[
     OrdinaryLinearProgrammingSolution[[n, k]]/k^(s), {k, 2, n}], {n,
     1, nn}];, n]

Monitor[lowerBound4 =
   Table[Sum[
     Sign[OrdinaryLinearProgrammingSolution[[n, k]]], {k, 2, n}], {n,
     1, nn}];, n]
Count[Table[Floor[(Sqrt[1 + 8*(n + 1)] - 3)/2], {n, 1, nn}] +
  lowerBound4, 0]

Show[ListLinePlot[constant*lowerBound3, PlotStyle -> {Thick}],
 ListPlot[constant*lowerBound3, PlotStyle -> {Thick},
  PlotMarkers -> Automatic],
 ListPlot[lowerBound1, PlotStyle -> {Thick, Red},
  PlotMarkers -> Automatic],
 ListLinePlot[lowerBound1, PlotStyle -> {Thick, Red}],
 ListPlot[constant*lowerBound2, PlotStyle -> {Thick},
  PlotMarkers -> Automatic],
 ListLinePlot[constant*lowerBound2, PlotStyle -> {Thick},
  PlotMarkers -> Automatic], ImageSize -> Large]
(*end*)