## 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]```