Fourier transform of exponential sawtooth and Riemann zeta function on the critical line

(*Mathematica 8*)
Clear[f]
scale = 1000000;
f = Range[scale] + 1;
xres = .002;
xlist = Exp[Range[0, Log[scale], xres]];
tmax = 60;
tres = .015;
Monitor[errList =
Table[((xlist^(1/2 - 1 + I t).(f[[Floor[xlist]]] - xlist)))*(1/2 +
I t), {t, Range[0, 60, tres]}];, t]
ListLinePlot[-Re[errList]/Length[xlist], DataRange -> {0, 60},
PlotRange -> {-0.1, .3}, Axes -> True, Filling -> Axis]

Plot[Re[(Zeta[1/2 + I t])], {t, 0, 60}, Filling -> Axis]

Clear[f]
scale = 1000000;
f = Range[scale] - 1;
xres = .002;
xlist = Exp[Range[0, Log[scale], xres]];
tmax = 60;
tres = .015;
Monitor[errList =
Table[((xlist^(1/2 - 1 + I t).(f[[Floor[xlist]]] - xlist)))*(1/2 +
I t), {t, Range[0, 60, tres]}];, t]
ListLinePlot[Im[errList]/Length[xlist], DataRange -> {0, 60},
PlotRange -> {-.2, .2}, Axes -> True, Filling -> Axis]


Plot[Im[(Zeta[1/2 + I t])], {t, 0, 60}, Filling -> Axis]

Clear[f]
scale = 1000000;
f = Range[scale];
xres = .002;
xlist = Exp[Range[0, Log[scale], xres]];
tmax = 60;
tres = .015;
Monitor[errList1 =
Table[((xlist^(1/2 - 1 + I t).(f[[Floor[xlist]]] - xlist + 1)))*(1/
2 + I t), {t, Range[0, 60, tres]}];, t]
Monitor[errList2 =
Table[((xlist^(1/2 - 1 + I t).(f[[Floor[xlist]]] - xlist - 1)))*(1/
2 + I t), {t, Range[0, 60, tres]}];, t]
ListLinePlot[{-Re[errList1]/Length[xlist],
Im[errList2]/Length[xlist]}, DataRange -> {0, 60},
PlotRange -> {-.2, .3}, Axes -> True, Filling -> Axis]

Plot[{Re[(Zeta[1/2 + I t])], Im[(Zeta[1/2 + I t])]}, {t, 0, 60},
Filling -> Axis]

Clear[f]
scale = 1000000;
f = Range[scale];
xres = .002;
xlist = Exp[Range[0, Log[scale], xres]];
tmax = 60;
tres = .015;
Monitor[errList1 =
Table[((xlist^(1/2 - 1 + I t).(SawtoothWave[xlist] - 1)))*(1/2 +
I t), {t, Range[0, 60, tres]}];, t]
Monitor[errList2 =
Table[((xlist^(1/2 - 1 + I t).(SawtoothWave[xlist] + 1)))*(1/2 +
I t), {t, Range[0, 60, tres]}];, t]
ListLinePlot[{Re[errList1]/Length[xlist], -Im[errList2]/
Length[xlist]}, DataRange -> {0, 60}, PlotRange -> {-.2, .3},
Axes -> True, Filling -> Axis]

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One Response to Fourier transform of exponential sawtooth and Riemann zeta function on the critical line

  1. Pingback: Are the amplitudes of these frequency spikes equal to 1 when the real part of the complex number “s” is equal to one half? - MathHub

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