The Mobius function Blog

Magic series and Magic constants

Posted on May 16, 2013 by Mats Granvik

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*)

Craig-Knecht-Magic-Series 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*)

Craig-Knecht-Magic-Series with partitions containing duplicates deleted Magic Series

Posted in Uncategorized

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

Posted on January 19, 2013 by Mats Granvik

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]

Posted in Uncategorized

The summation symbol

Posted on January 12, 2013 by Mats Granvik

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

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

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

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

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

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

Posted in Uncategorized

A visual interpretation of Riemann zeta zeros via the Fourier transform

Posted on December 18, 2012 by Mats Granvik

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.

Posted in Uncategorized

Illustration of the Discrete Fourier Tranform DFT

Posted on December 17, 2012 by Mats Granvik

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

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

Posted in Uncategorized

Dirichlet character sums for the terms of the von Mangoldt function

Posted on November 4, 2012 by Mats Granvik

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

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

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

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

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

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

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

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

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

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

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

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

Posted in Uncategorized

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

Posted on October 13, 2012 by Mats Granvik

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.

Posted in Uncategorized

	Mats Granvik on The inverse of a triangular ma…
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The Mobius function Blog

Magic series and Magic constants

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

The summation symbol

A visual interpretation of Riemann zeta zeros via the Fourier transform

Illustration of the Discrete Fourier Tranform DFT

Dirichlet character sums for the terms of the von Mangoldt function

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

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