progressive spectra

Clear[n, k, t, A, nn]
nnn = 12;
Show[Flatten[{Table[
    ListLinePlot[
     Table[Re[
       Total[1/Table[n*t, {n, 1, nn}]*
         Total[Transpose[
           Inverse[
             Table[Table[
               If[Mod[n, k] == 0, N[1/(n/k)^(1/2 + I*t - 1)], 0], {k,
                1, nn}], {n, 1, nn}]]*Zeta[1/2 + I*t]]]]], {t, 1/1000,
        60, N[1/10]}], DataRange -> {0, 60},
     PlotRange -> {-0.15, 1.5}], {nn, 1, nnn}],
   Table[ListLinePlot[
     Table[Re[
       Total[1/Table[n*t, {n, 1, nn}]*
         Total[Transpose[
           Inverse[
             Table[Table[
               If[Mod[n, k] == 0, N[1/(n/k)^(1/2 + I*t - 1)], 0], {k,
                1, nn}], {n, 1, nn}]]*Zeta[1/2 + I*t]]]]], {t, 1/1000,
        60, N[1/10]}], DataRange -> {0, 60},
     PlotRange -> {-0.15, 1.5}, PlotStyle -> Red], {nn, nnn, nnn}]}]]