## Local minima of Fourier transform of natural numbers close to imaginary parts of Riemann zeta zeros

Posted to math stackexchange:
What are the exact values of the local minima in this spectrum?

Apply this program, originally written by Heike and here modified:

```(*Mathematica 8 start*) Clear[f] scale = 1000000; f = Range[scale] - 1; xres = .002; xlist = Exp[Range[0, Log[scale], xres]]; tmax = 100; tres = .015; ymin = -0.025; ymax = 0.015; zz = N[Im[ZetaZero[1]]]; Monitor[errList = Table[(xlist^(-1/2 + I t).(f[[Floor[xlist]]] - xlist)), {t, Range[0, tmax, tres]}];, t] g1 = ListLinePlot[Im[errList]/Length[xlist], DataRange -> {0, tmax}, PlotRange -> {ymin, ymax}, Frame -> True, Axes -> False]; g2 = Graphics[ Line[{{N[Im[ZetaZero[1]]], ymin}, {N[Im[ZetaZero[1]]], ymax}}]]; g3 = Graphics[ Line[{{N[Im[ZetaZero[2]]], ymin}, {N[Im[ZetaZero[2]]], ymax}}]]; g4 = Graphics[ Line[{{N[Im[ZetaZero[3]]], ymin}, {N[Im[ZetaZero[3]]], ymax}}]]; g5 = Graphics[ Line[{{N[Im[ZetaZero[4]]], ymin}, {N[Im[ZetaZero[4]]], ymax}}]]; g6 = Graphics[ Line[{{N[Im[ZetaZero[5]]], ymin}, {N[Im[ZetaZero[5]]], ymax}}]]; g7 = Graphics[ Line[{{N[Im[ZetaZero[6]]], ymin}, {N[Im[ZetaZero[6]]], ymax}}]]; g8 = Graphics[ Line[{{N[Im[ZetaZero[7]]], ymin}, {N[Im[ZetaZero[7]]], ymax}}]]; g9 = Graphics[ Line[{{N[Im[ZetaZero[8]]], ymin}, {N[Im[ZetaZero[8]]], ymax}}]]; g10 = Graphics[ Line[{{N[Im[ZetaZero[9]]], ymin}, {N[Im[ZetaZero[9]]], ymax}}]]; g11 = Graphics[ Line[{{N[Im[ZetaZero[10]]], ymin}, {N[Im[ZetaZero[10]]], ymax}}]]; g12 = Graphics[ Line[{{N[Im[ZetaZero[11]]], ymin}, {N[Im[ZetaZero[11]]], ymax}}]]; g13 = Graphics[ Line[{{N[Im[ZetaZero[12]]], ymin}, {N[Im[ZetaZero[12]]], ymax}}]]; g14 = Graphics[ Line[{{N[Im[ZetaZero[13]]], ymin}, {N[Im[ZetaZero[13]]], ymax}}]]; g15 = Graphics[ Line[{{N[Im[ZetaZero[14]]], ymin}, {N[Im[ZetaZero[14]]], ymax}}]]; g16 = Graphics[ Line[{{N[Im[ZetaZero[15]]], ymin}, {N[Im[ZetaZero[15]]], ymax}}]]; g17 = Graphics[ Line[{{N[Im[ZetaZero[16]]], ymin}, {N[Im[ZetaZero[16]]], ymax}}]]; g18 = Graphics[ Line[{{N[Im[ZetaZero[17]]], ymin}, {N[Im[ZetaZero[17]]], ymax}}]]; g19 = Graphics[ Line[{{N[Im[ZetaZero[18]]], ymin}, {N[Im[ZetaZero[18]]], ymax}}]]; g20 = Graphics[ Line[{{N[Im[ZetaZero[19]]], ymin}, {N[Im[ZetaZero[19]]], ymax}}]]; g21 = Graphics[ Line[{{N[Im[ZetaZero[20]]], ymin}, {N[Im[ZetaZero[20]]], ymax}}]]; g22 = Graphics[ Line[{{N[Im[ZetaZero[21]]], ymin}, {N[Im[ZetaZero[21]]], ymax}}]]; g23 = Graphics[ Line[{{N[Im[ZetaZero[22]]], ymin}, {N[Im[ZetaZero[22]]], ymax}}]]; g24 = Graphics[ Line[{{N[Im[ZetaZero[23]]], ymin}, {N[Im[ZetaZero[23]]], ymax}}]]; g25 = Graphics[ Line[{{N[Im[ZetaZero[24]]], ymin}, {N[Im[ZetaZero[24]]], ymax}}]]; g26 = Graphics[ Line[{{N[Im[ZetaZero[25]]], ymin}, {N[Im[ZetaZero[25]]], ymax}}]]; g27 = Graphics[ Line[{{N[Im[ZetaZero[26]]], ymin}, {N[Im[ZetaZero[26]]], ymax}}]]; g28 = Graphics[ Line[{{N[Im[ZetaZero[27]]], ymin}, {N[Im[ZetaZero[27]]], ymax}}]]; g29 = Graphics[ Line[{{N[Im[ZetaZero[28]]], ymin}, {N[Im[ZetaZero[28]]], ymax}}]]; g30 = Graphics[ Line[{{N[Im[ZetaZero[29]]], ymin}, {N[Im[ZetaZero[29]]], ymax}}]]; gw = Show[g1, g2, g3, g4, g5, g6, g7, g8, g9, g10, g11, g12, g13, g14, g15, g16, g17, g18, g19, g20, g21, g22, g23, g24, g25, g26, g27, g28, g29, g30, ImageSize -> Large] (*Mathematica 8 end*)```

The result is a plot with local minima that are close to the imaginary part of the Riemann zeta zeros: