Riemann zeta function on the critical line as a Fourier transform of exponential sawtooth function plus minus one

Riemann zeta function on the critical line as a Fourier transform of exponential sawtooth function plus minus one

Riemann zeta function on critical line

The code does not work when copy pasted in this blogging platform,
so here is a link to Pastebin with some working code:

http://pastebin.com/TC1wcuzF

Mathematica:

scale = 1000000;
xres = .00001;
x = Exp[Range[0, Log[scale], xres]];
RealPart = Log[x]*FourierDST[-(SawtoothWave[x] – 1)*x^(-1/2)];
ImaginaryPart = Log[x]*FourierDCT[-(SawtoothWave[x] + 1)*x^(-1/2)];
datapointsdisplayed = 300;
ymin = -15;
ymax = 15;
g1 = ListLinePlot[
Sqrt[scale]*{RealPart[[1 ;; datapointsdisplayed]],
ImaginaryPart[[1 ;; datapointsdisplayed]]},
PlotRange -> {ymin, ymax}, DataRange -> {0, 68.00226987379779},
Filling -> Axis];
Show[Flatten[{g1,
Table[Graphics[{PointSize[0.013],
Point[{N[Im[ZetaZero[n]]], 0}]}], {n, 1, 16}]}],
ImageSize -> Large]

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The summation symbol

\sum\limits_{n=1}^1 1=1

\sum\limits_{n=1}^2 1=1+1

\sum\limits_{n=1}^3 1=1+1+1

\sum\limits_{n=1}^4 1=1+1+1+1

\sum\limits_{n=1}^5 1=1+1+1+1+1

\sum\limits_{n=1}^6 1=1+1+1+1+1+1

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A visual interpretation of Riemann zeta zeros via the Fourier transform

Mathematica 8:

scale = 1000000;
xres = .001;
limit = 3000;
x = Exp[Range[0, Log[scale], xres]];
a = FourierDCT[(SawtoothWave[x])*x^(-1/2)];
b = -FourierDST[(SawtoothWave[x] – 1)*x^(-1/2)];
(*ListLinePlot[((SawtoothWave[x])*x^(-1/2))[[1;;limit]]]*)
gs = ListLinePlot[-((SawtoothWave[x] – 1)*x^(-1/2))[[1 ;; limit]],
PlotStyle -> RGBColor[1, 0, 1]];
gsine = ListLinePlot[
Table[Sin[Im[ZetaZero[1]] x], {x, 0, limit*xres, xres}]];
Show[gs, gsine, PlotRange -> {-1, 1}]
ListLinePlot[
Table[Sin[
Im[ZetaZero[1]] x]*(-((SawtoothWave[x] – 1)*x^(-1/2))), {x,
0.00001, limit*xres, xres}]]

Frequency interpretation of Riemann zeta zero via the Fourier transform:

Frequency interpretation of Riemann zeta zeros

The two curves multiplied:

Frequency interpretation of Riemann zeta zeros - the two functions multiplied

I believe the sum of the latter should be zero in order for the
frequency of the sine curve to be equal to a Riemann zeta zero.

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Illustration of the Discrete Fourier Tranform DFT

Mathematica 8:

Do[
nn = i;
Print[MatrixForm[Transpose[Table[{1}, {n, 1, nn}]]]]
Print[MatrixForm[
Table[Table[Cos[-2*Pi*(n – 1)*(k – 1)/nn], {k, 1, nn}], {n, 1,
nn}]]]
Print[MatrixForm[
Chop[N[Table[
Total[Table[Cos[-2*Pi*(n – 1)*(k – 1)/nn], {k, 1, nn}]], {n, 1,
nn}]]]]]
, {i, 1, 12}]

Signal or time domain

\left(  \begin{array}{c}   1  \end{array}  \right)

Dicrete Fourier Cosine Transform

\left(  \begin{array}{c}   1  \end{array}  \right)

Spectrum or frequency domain

\left(  \begin{array}{c}   1.  \end{array}  \right)

Signal or time domain

\left(  \begin{array}{cc}   1 & 1  \end{array}  \right)

Dicrete Fourier Cosine Transform

\left(  \begin{array}{cc}   1 & 1 \\   1 & -1  \end{array}  \right)

Spectrum or frequency domain

\left(  \begin{array}{c}   2. \\   0.  \end{array}  \right)

Signal or time domain

\left(  \begin{array}{ccc}   1 & 1 & 1  \end{array}  \right)

Dicrete Fourier Cosine Transform

\left(  \begin{array}{ccc}   1 & 1 & 1 \\   1 & -\frac{1}{2} & -\frac{1}{2} \\   1 & -\frac{1}{2} & -\frac{1}{2}  \end{array}  \right)

Spectrum or frequency domain

\left(  \begin{array}{c}   3. \\   0. \\   0.  \end{array}  \right)

Signal or time domain

\left(  \begin{array}{cccc}   1 & 1 & 1 & 1  \end{array}  \right)

Dicrete Fourier Cosine Transform

\left(  \begin{array}{cccc}   1 & 1 & 1 & 1 \\   1 & 0 & -1 & 0 \\   1 & -1 & 1 & -1 \\   1 & 0 & -1 & 0  \end{array}  \right)

Spectrum or frequency domain

\left(  \begin{array}{c}   4. \\   0. \\   0. \\   0.  \end{array}  \right)

Signal or time domain

\left(  \begin{array}{ccccc}   1 & 1 & 1 & 1 & 1  \end{array}  \right)

Dicrete Fourier Cosine Transform

\left(  \begin{array}{ccccc}   1 & 1 & 1 & 1 & 1 \\   1 & \frac{1}{4} \left(-1+\sqrt{5}\right) & \frac{1}{4} \left(-1-\sqrt{5}\right) & \frac{1}{4} \left(-1-\sqrt{5}\right) & \frac{1}{4} \left(-1+\sqrt{5}\right) \\   1 & \frac{1}{4} \left(-1-\sqrt{5}\right) & \frac{1}{4} \left(-1+\sqrt{5}\right) & \frac{1}{4} \left(-1+\sqrt{5}\right) & \frac{1}{4} \left(-1-\sqrt{5}\right) \\   1 & \frac{1}{4} \left(-1-\sqrt{5}\right) & \frac{1}{4} \left(-1+\sqrt{5}\right) & \frac{1}{4} \left(-1+\sqrt{5}\right) & \frac{1}{4} \left(-1-\sqrt{5}\right) \\   1 & \frac{1}{4} \left(-1+\sqrt{5}\right) & \frac{1}{4} \left(-1-\sqrt{5}\right) & \frac{1}{4} \left(-1-\sqrt{5}\right) & \frac{1}{4} \left(-1+\sqrt{5}\right)  \end{array}  \right)

Spectrum or frequency domain

\left(  \begin{array}{c}   5. \\   0. \\   0. \\   0. \\   0.  \end{array}  \right)

Signal or time domain

\left(  \begin{array}{cccccc}   1 & 1 & 1 & 1 & 1 & 1  \end{array}  \right)

Dicrete Fourier Cosine Transform

\left(  \begin{array}{cccccc}   1 & 1 & 1 & 1 & 1 & 1 \\   1 & \frac{1}{2} & -\frac{1}{2} & -1 & -\frac{1}{2} & \frac{1}{2} \\   1 & -\frac{1}{2} & -\frac{1}{2} & 1 & -\frac{1}{2} & -\frac{1}{2} \\   1 & -1 & 1 & -1 & 1 & -1 \\   1 & -\frac{1}{2} & -\frac{1}{2} & 1 & -\frac{1}{2} & -\frac{1}{2} \\   1 & \frac{1}{2} & -\frac{1}{2} & -1 & -\frac{1}{2} & \frac{1}{2}  \end{array}  \right)

Spectrum or frequency domain

\left(  \begin{array}{c}   6. \\   0. \\   0. \\   0. \\   0. \\   0.  \end{array}  \right)

Signal or time domain

\left(  \begin{array}{ccccccc}   1 & 1 & 1 & 1 & 1 & 1 & 1  \end{array}  \right)

Dicrete Fourier Cosine Transform

\left(  \begin{array}{ccccccc}   1 & 1 & 1 & 1 & 1 & 1 & 1 \\   1 & \text{Sin}\left[\frac{3 \pi }{14}\right] & -\text{Sin}\left[\frac{\pi }{14}\right] & -\text{Cos}\left[\frac{\pi }{7}\right] & -\text{Cos}\left[\frac{\pi }{7}\right] & -\text{Sin}\left[\frac{\pi }{14}\right] & \text{Sin}\left[\frac{3 \pi }{14}\right] \\   1 & -\text{Sin}\left[\frac{\pi }{14}\right] & -\text{Cos}\left[\frac{\pi }{7}\right] & \text{Sin}\left[\frac{3 \pi }{14}\right] & \text{Sin}\left[\frac{3 \pi }{14}\right] & -\text{Cos}\left[\frac{\pi }{7}\right] & -\text{Sin}\left[\frac{\pi }{14}\right] \\   1 & -\text{Cos}\left[\frac{\pi }{7}\right] & \text{Sin}\left[\frac{3 \pi }{14}\right] & -\text{Sin}\left[\frac{\pi }{14}\right] & -\text{Sin}\left[\frac{\pi }{14}\right] & \text{Sin}\left[\frac{3 \pi }{14}\right] & -\text{Cos}\left[\frac{\pi }{7}\right] \\   1 & -\text{Cos}\left[\frac{\pi }{7}\right] & \text{Sin}\left[\frac{3 \pi }{14}\right] & -\text{Sin}\left[\frac{\pi }{14}\right] & -\text{Sin}\left[\frac{\pi }{14}\right] & \text{Sin}\left[\frac{3 \pi }{14}\right] & -\text{Cos}\left[\frac{\pi }{7}\right] \\   1 & -\text{Sin}\left[\frac{\pi }{14}\right] & -\text{Cos}\left[\frac{\pi }{7}\right] & \text{Sin}\left[\frac{3 \pi }{14}\right] & \text{Sin}\left[\frac{3 \pi }{14}\right] & -\text{Cos}\left[\frac{\pi }{7}\right] & -\text{Sin}\left[\frac{\pi }{14}\right] \\   1 & \text{Sin}\left[\frac{3 \pi }{14}\right] & -\text{Sin}\left[\frac{\pi }{14}\right] & -\text{Cos}\left[\frac{\pi }{7}\right] & -\text{Cos}\left[\frac{\pi }{7}\right] & -\text{Sin}\left[\frac{\pi }{14}\right] & \text{Sin}\left[\frac{3 \pi }{14}\right]  \end{array}  \right)

Spectrum or frequency domain

\left(  \begin{array}{c}   7. \\   0 \\   0 \\   0 \\   0 \\   0 \\   0  \end{array}  \right)

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Dirichlet character sums for the terms of the von Mangoldt function

\sum _{n=1}^{\infty } (1 \chi _{1,1}(n)+0)

\log(2) = \sum _{n=1}^{\infty } (2 \chi _{2,1}(n)-1)

\log(3) = \sum _{n=1}^{\infty } (3 \chi _{3,1}(n)-2)

\log(2) = \sum _{n=1}^{\infty } (2 \chi _{4,1}(n)-1)

\log(5) = \sum _{n=1}^{\infty } (5 \chi _{5,1}(n)-4)

\log(1) = \sum _{n=1}^{\infty } (2 \chi _{2,1}(n)-1) (3 \chi _{3,1}(n)-2)

\log(7) = \sum _{n=1}^{\infty } (7 \chi _{7,1}(n)-6)

\log(2) = \sum _{n=1}^{\infty } (2 \chi _{8,1}(n)-1)

\log(3) = \sum _{n=1}^{\infty } (3 \chi _{9,1}(n)-2)

\log(1) = \sum _{n=1}^{\infty } (2 \chi _{2,1}(n)-1) (5 \chi _{5,1}(n)-4)

\log(11) = \sum _{n=1}^{\infty } (11 \chi _{11,1}(n)-10)

\log(1) = \sum _{n=1}^{\infty } (4 \chi _{4,1}(n)-1) (3 \chi _{3,1}(n)-2)

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The fundamental theorem of arithmetic is encoded by the von Mangoldt function

Mathematica 8

A = Table[
Table[If[Mod[n, k] == 0, Exp[MangoldtLambda[n/k]], ""], {k, 1,
12}], {n, 1, 12}];
MatrixForm[A]

\left(  \begin{array}{cccccccccccc}   1 & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} \\   2 & 1 & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} \\   3 & \text{} & 1 & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} \\   2 & 2 & \text{} & 1 & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} \\   5 & \text{} & \text{} & \text{} & 1 & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} \\   1 & 3 & 2 & \text{} & \text{} & 1 & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} \\   7 & \text{} & \text{} & \text{} & \text{} & \text{} & 1 & \text{} & \text{} & \text{} & \text{} & \text{} \\   2 & 2 & \text{} & 2 & \text{} & \text{} & \text{} & 1 & \text{} & \text{} & \text{} & \text{} \\   3 & \text{} & 3 & \text{} & \text{} & \text{} & \text{} & \text{} & 1 & \text{} & \text{} & \text{} \\   1 & 5 & \text{} & \text{} & 2 & \text{} & \text{} & \text{} & \text{} & 1 & \text{} & \text{} \\   11 & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & 1 & \text{} \\   1 & 1 & 2 & 3 & \text{} & 2 & \text{} & \text{} & \text{} & \text{} & \text{} & 1  \end{array}  \right)

Row products of the matrix above are the natural numbers.

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Agreement between summatory von Mangoldt function and partial sums of von Mangoldt matrix

Edit 14.10.2012: Unfortunately copy pasting into wordpress makes the code show wrong, and it will not work.

Mathematica 8

aa = 32;
a = Range[aa]*0;
Monitor[Do[
T[n_, k_] :=
T[n, k] =
If[n < 1 || k n, T[k, n],
If[n > k, T[k, Mod[n, k, 1]], -Sum[T[n, i], {i, n – 1}]]]]];
A = Table[Table[T[n, k]/n, {n, 1, nn}], {k, 1, nn}];
A[[1, All]] = 0;
a[[nn]] = Total[Total[A]], {nn, 1, aa}], nn]
b = a;
c = Accumulate[Table[N[MangoldtLambda[n]], {n, 1, aa}]];
g2 = ListPlot[{c, b}, ImageSize -> Full]

Link to Pastebin with working code:
von Mangoldt function and von Mangoldt matrix

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