Shifted spikes in Fourier spectra of primes.

Clear[n, k, t, A, nn, B]
b = {1}
nn = 60
A = Table[
   Table[If[Mod[n, k] == 0, 1/(n/k)^(1/2 + I*t - 1), 0], {k, 1,
     nn}], {n, 1, nn}];
MatrixForm[A];
inverseA = Inverse[A];
inverseA =
  Table[Table[
    inverseA[[n, k]] b[[1 + Mod[n - 1, 1]]], {k, 1, nn}], {n, 1, nn}];
B = FourierDCT[
   Table[Total[
     1/Table[n, {n, 1, nn}]*(Total[
         Transpose[Re[inverseA*Zeta[1/2 + I*t]]]] - 1)], {t, 1/1000,
     600, N[1/6]}]];
g1 = ListLinePlot[B[[1 ;; 700]], DataRange -> {0, 60},
   PlotRange -> {-10, 30}, Axes -> False];
mm = 11.35/Log[2];
g2 = Graphics[
   Table[Style[Text[n, {mm*Log[n], 5 - (-1)^n}],
     FontFamily -> "Times New Roman", FontSize -> 14], {n, 1, 16}]];
Show[g1, g2, ImageSize -> Large]


Clear[n, k, t, A, nn, B]
b = {1, -1}
nn = 60
A = Table[
   Table[If[Mod[n, k] == 0, 1/(n/k)^(1/2 + I*t - 1), 0], {k, 1,
     nn}], {n, 1, nn}];
MatrixForm[A];
inverseA = Inverse[A];
inverseA =
  Table[Table[
    inverseA[[n, k]] b[[1 + Mod[n - 1, 2]]], {k, 1, nn}], {n, 1, nn}];
B = FourierDCT[
   Table[Total[
     1/Table[n, {n, 1, nn}]*(Total[
         Transpose[Re[inverseA*Zeta[1/2 + I*t]]]] - 1)], {t, 1/1000,
     600, N[1/6]}]];
g1 = ListLinePlot[B[[1 ;; 700]], DataRange -> {0, 60},
   PlotRange -> {-10, 60}, Axes -> False];
mm = 11.35/Log[2];
g2 = Graphics[
   Table[Style[Text[n, {mm*Log[n], 5 - (-1)^n}],
     FontFamily -> "Times New Roman", FontSize -> 14], {n, 1, 16}]];
Show[g1, g2, ImageSize -> Large]


Clear[n, k, t, A, nn, B]
b = {1, 1, -2}
nn = 60
A = Table[
   Table[If[Mod[n, k] == 0, 1/(n/k)^(1/2 + I*t - 1), 0], {k, 1,
     nn}], {n, 1, nn}];
MatrixForm[A];
inverseA = Inverse[A];
inverseA =
  Table[Table[
    inverseA[[n, k]] b[[1 + Mod[n - 1, 3]]], {k, 1, nn}], {n, 1, nn}];
B = FourierDCT[
   Table[Total[
     1/Table[n, {n, 1, nn}]*(Total[
         Transpose[Re[inverseA*Zeta[1/2 + I*t]]]] - 1)], {t, 1/1000,
     600, N[1/6]}]];
g1 = ListLinePlot[B[[1 ;; 700]], DataRange -> {0, 60},
   PlotRange -> {-10, 60}, Axes -> False];
mm = 11.35/Log[2];
g2 = Graphics[
   Table[Style[Text[n, {mm*Log[n], 5 - (-1)^n}],
     FontFamily -> "Times New Roman", FontSize -> 14], {n, 1, 16}]];
Show[g1, g2, ImageSize -> Large]