Dirichlet series for a symmetric matrix

Let \mu(n) be the Möbius function

a(n) = \sum\limits_{d|n} d \cdot \mu(d)

T(n,k)=a(GCD(n,k))

\sum\limits_{k=1}^{\infty}\sum\limits_{n=1}^{\infty} \frac{T(n,k)}{n^c \cdot k^s} = \sum\limits_{n=1}^{\infty} \frac{\lim\limits_{z \rightarrow s} \zeta(s)\sum\limits_{d|n} \frac{\mu(d)}{d^{(z-1)}}}{n^c} = \frac{\zeta(s) \cdot \zeta(c)}{\zeta(c + s - 1)}

-\frac{\zeta '(s)}{\zeta (s)}=\lim_{c\to 1} \, \left(\frac{\zeta (c) \zeta (s)}{\zeta (c+s-1)}-\zeta (c)\right)

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Train of thought leading from the zeta function to the Möbius function

(*start Mathematica 8*)
(*Start with Riemann zeta:*)
Zeta[s]
(*Take the logarithm:*)
Log[Zeta[s]]
(*Take the derivative:*)
D[Log[Zeta[s]], s]
Clear[s, c]
(*Generalize it:*)
Limit[Zeta[c] - Zeta[s]*Zeta[c]/Zeta[s + c - 1], c -> 1]
(*See that Zeta[s]*Zeta[c]/Zeta[s+c-1] is the Dirichlet generating \
function of:*)
Table[Limit[
Zeta[s]*Total[MoebiusMu[Divisors[n]]/Divisors[n]^(s - 1)]/n^c,
s -> 1], {n, 1, 12}]
(*Which in turn is the Dirichlet generating function of the rows or \
columns of the symmetric matrix:*)
nn = 32;
A = Table[
Table[If[Mod[n, k] == 0, k^ZetaZero[k], 0], {k, 1, nn}], {n, 1,
nn}];
B = Table[
Table[If[Mod[k, n] == 0, MoebiusMu[n]*n^ZetaZero[-n], 0], {k, 1,
nn}], {n, 1, nn}];
MatrixForm[N[A.B]]
(*For comparison,here is a plot of the von Mangoldt function from the \
matrix:*)
ListLinePlot[Total[N[A.B]/Range[nn]]]
(*end*)

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The Möbius function times n

1, -2, -3, 0, -5, 6, -7, 0, 0, 10, -11, 0, -13, 14, 15, 0, -17, 0, -19, 0, 21, 22, -23, 0, 0, 26, 0, 0, -29, -30, -31, 0, 33, 34, 35, 0, -37, 38, 39, 0, -41, -42, -43, 0, 0, 46, -47, 0, 0, 0, 51, 0, -53, 0, 55, 0, 57, 58, -59, 0, -61, 62, 0, 0, 65, -66, -67, 0, 69, -70, -71, 0, -73, 74, 0, 0, 77, -78, -79, 0, 0, 82, -83, 0, 85, 86, 87, 0, -89, 0, 91, 0, 93, 94, 95, 0, -97, 0, 0, 0

Mathematica: MoebiusMu[Range[100]]*Range[100]

Keywords: core, sign, number theory

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Arne Bergstroms paper 26 6 2013

u=2 i \pi  c_2+\log \left(i \left(2 \pi  c_1+\pi \right)\right)

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Arne Bergstroms paper

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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*)

Craig-Knecht-Magic-SeriesInteger 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 deletedMagic 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

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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