       Re: Problem with Infinite products--> further modulo 12 prime functions

• To: mathgroup at smc.vnet.net
• Subject: [mg65462] Re: Problem with Infinite products--> further modulo 12 prime functions
• From: Roger Bagula <rlbagulatftn at yahoo.com>
• Date: Mon, 3 Apr 2006 06:59:27 -0400 (EDT)
• References: <dvrbsp\$a3a\$1@smc.vnet.net> <e035oq\$2d8\$1@smc.vnet.net> <e05re8\$3r5\$1@smc.vnet.net> <e08kk8\$4b1\$1@smc.vnet.net> <e0dp0n\$q53\$1@smc.vnet.net> <e0gco3\$i5s\$1@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

```Yes, I seem to have come up with a way to factor the Zeta function into
five parts. It also seem to work on other functions.

Using: "n_Integer?Positive" gets rid of the error signal too.
Here are harmonic functions "deconstructed" into nearly orthogonal
sines and cosines.

f[n_Integer?Positive, 1] := If[Mod[Prime[n], 12] - 1 == 0, n!, 0]
f[n_Integer?Positive, 2] := If[Mod[Prime[n], 12] - 5 == 0, n!, 0]
f[n_Integer?Positive, 3] := If[Mod[Prime[n], 12] - 7 == 0, n!, 0]
f[n_Integer?Positive, 4] := If[Mod[Prime[n], 12] - 11 == 0, n!, 0]
CosM[x_, m_Integer?Positive] := Sum[If[f[2*n, m] == 0,
0, (-1)^n*x^(2*n)/f[2*n, m]], {n, 1, 200}]
SinM[x_, m_Integer?Positive] := Sum[If[f[2*n + 1, m] == 0, 0, (-1)^n*x^(2*
n + 1)/f[2*n + 1, m]], {n, 1, 200}]
ac = Table[Plot[CosM[x, m], {x, -Pi, Pi}, PlotRange -> All], {m, 1, 4}]
as = Table[Plot[SinM[x, m], {x, -Pi, Pi}, PlotRange -> All], {m, 1, 4}]

This result gave me the idea that maybe the the primes were behaving
like polarized light. Here is my experiment of this morning in that
"light": ( you will note the orthogonality condition holds:
ai[n]*aj[n]=0 i,j={1,5,7,11}  )

(* appearance functional vectors for Primes Mudulo 12*)
a1[n_Integer?Positive] := If[Mod[Prime[n], 12] - 1 == 0, 1, 0]
a5[n_Integer?Positive] := If[Mod[Prime[n], 12] - 5 == 0, 1, 0]
a7[n_Integer?Positive] := If[Mod[Prime[n], 12] - 7 == 0, 1, 0]
a11[n_Integer?Positive] := If[Mod[Prime[n], 12] - 11 == 0, 1, 0]
(*Phase vectors relative to primes modulo 12 ending in one*)
p5[n_Integer?Positive] := If[a1[n] - 1 == 0,
a1[n]*a5[n + 1] + a1[n]*a5[n + 2] + a1[n]*a5[n + 3], 0]
p7[n_Integer?Positive] := If[a1[n] - 1 == 0, a1[n]*a7[n + 1] + a1[n]*a7[
n + 2] + a1[n]*a7[n + 3], 0]
p11[n_Integer?Positive] := If[a1[n] -
1 == 0, a1[n]*a11[n + 1] + a1[n]*a11[n + 2] + a1[n]*a11[n + 3], 0]
(* Average phase functions *)
PA5[m_Integer?Positive] := (4/m)*Sum[a1[n]*p5[n], {n, 1.m}]
PA7[m_Integer?Positive] := (4/m)*Sum[a1[n]*p7[n], {n, 1.m}]
PA11[m_Integer?Positive] := (4/m)*Sum[a1[n]*p11[n], {n, 1.m}]
Digits = 1000
a = Table[PA5[n], {n, 1, Digits}];
ListPlot[a]
b = Table[PA7[n], {n, 1, Digits}];
ListPlot[b]
c = Table[PA11[n], {n, 1, Digits}];
ListPlot[c]
w = Table[Point[{a[[n]], b[[n]], c[[n]]}], {n, 1, Digits}];
Show[Graphics3D[{PointSize[0.005], w}]]

In this experiment after a large number it becomes apparent that 5 and
11 ending modulo 12 primes behave at very close to the same phase angle.
I had done other experiments after finding that even and odd primes
behaved differently in modulo 12 that showed a pairing effect of types
{1,7},{5,11}
So that you get complex pairs of Sines and Cosines like :
C1+I*S5
C7+I*S11
Using the above functions:
x0 := 10*CosM[t, 1]*CosM[p, 3];
y0 := SinM[t, 2]*CosM[p, 3]
z0 := SinM[p, 4]
ParametricPlot3D[{z0, y0, x0}, {t, -Pi, Pi}, {p, -Pi,
Pi}, PlotPoints -> {40, 40}, Boxed -> False, Axes -> False]

Which appears to be an interesting new surface.

It is interesting that such n! factorial based function are most
dependent on the initial area where these  phases are most chaotic.
Maxim wrote:
> On Wed, 29 Mar 2006 10:54:15 +0000 (UTC), Roger Bagula

>
> Maxim Rytin
> m.r at inbox.ru
>

```

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