## Magic series and Magic constants

Craig Knecht sent me an email explaining magic series and magic constants. The following program lists magic series that add up to certain constants using the TableForm command in Mathematica:

Mathematica 8:

(*program for reordering of integer partitions start*)
TableForm[
Table[Table[
IntegerPartitions[
magicConstant][[Flatten[
Position[
Table[Length[IntegerPartitions[magicConstant][[i]]], {i, 1,
Length[IntegerPartitions[magicConstant]]}],
order]]]], {magicConstant, 1, 12}], {order, 1, 12}]]
(*program for reordering of integer partitions end*)

Integer Partitions

By removing the integer partitions that contain duplicates we get magic series:

Mathematica 8:

(*program for listing magic series start*)
nn = 14;
A = Table[
Table[IntegerPartitions[
magicConstant][[Flatten[
Position[
Table[Length[IntegerPartitions[magicConstant][[i]]], {i, 1,
Length[IntegerPartitions[magicConstant]]}],
order]]]], {magicConstant, 1, nn}], {order, 1, nn}];
TableForm[
Table[Table[
Table[If[
Length[DeleteDuplicates[A[[n]][[k]][[j]]]] ==
Length[A[[n]][[k]][[j]]], A[[n]][[k]][[j]], “”], {j, 1,
Length[A[[n]][[k]]]}], {n, 1, k}], {k, 1, nn}]]
(*program for listing magic series end*)

Magic Series

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## 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

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:

The two curves 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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