Abstract: The variable dimension of the Cantor set created by the sequence of the Primes contains a new kind of 1/f noise. In the investigation of this type of pattern I have created a new kind of dimensional self-similarity which I have named new chaos.

Body: At first look the primes seem just another sequence, but the greatest minds of mathematics have failed to ``crack'' the pattern or information involved in this sequence.

I don't claim in this article to have done any better, but I will introduce some tools I have invented for sets whose dimension changes dynamically as the set evolves. I have always used the computer as a way to help visualize mathematical problems. I read Reimann's paper on the primes and even this great man standing on the back of another great man Gauss couldn't get more than a chaotic estimate for how the primes evolve.

Let's look at the primes at some number n that may be arbitrary and large. The dimension in self-similar terms of the set is estimated by:
 

1) s(n)=Log(n)/Log(P(n))

Another way to look at the set is by the non-primes and their self-similar dimension which turns out to be larger than the first:
 

2) v(n)=Log(P(n)-n)/Log(P(n))

In both these cases the functions increase as n gets larger and in both cases they go toward a value of one. A variable like an expectation value is made using a sum:
 

3) S(n)=Sum[ log(i)/log(P(i)),{i,1, n}]/n

where the first prime is taken as 2. This kind of dimension is more what you see when the whole set is taken into account. A similar function can be made for the second type of dimension:
 

4) V(n)=Sum[ log(P(i)-i)/log(P(i)),{i,1, n}]/n

These dimensions are more like what a box counting or entropy dimension would give. Suppose we looked at a probability like entropy of the primes as:
 

5) h(n)=Sum[ Log(1/P(i))*(1/P(i)),{i,1,n}]

This function gets pretty large in absolute value terms, but another function is finite and limited:
 

6) H(x)=Sum[Sum[ (-1)^i*Log(1/P(i))*(1/P(i),{i,1,n}],{n,1,x}]/x

The results from this function seem to agree better with the two expectation value functions so that:
 

7) S(n)+H(n) ~ V(n)

The information seems to be varying in a very complex manner. Gauss and Riemann used a function that Gauss invented:
 

9) Li(x)=Integrate[1/log(n),{n,2,x}]

Reimann invented his own function with some very involved arguments:
 

10) F(x)=Li(x)-Sum[Li(x/n)/n,{n,2,x}]

This may be one of the earliest of a sum type of fractal function like the Weierstrass functions. I don't think Reimann considered scaling or Cantor sets or even knew about them.

In my work it became obvious that the information that the prime sequence contained was some sort of varying Cantor set where the dimension was low to begin with, but increased as the sequence increased. My first attempt at synthesis of such a function was an angular cyclic dimensional function:
 

11) w=angle(x,y)

12) s=(1-cos(w)/2 : Domain[0,1]

and a random switch between two transforms:
 

13) x'=(x/3)^s

14) x'=(x/3)^s+2/3

which I did in two dimensions with independent randoms. This did give a fractal, but not what I needed. I changed to a random iterative dimension and half based Cantor sets:
 

15) s'=s/2

16) s'=s/2+1/2

17) x'=(x/2)^s

18) x'=(x/2)+(1/2)^s
 

19) s'=s/2+1/s

20) s'=abs(s/2-1/2)

By using signed powers I was able to get Sierpinski, Eisenstein and twin dragon like

IFS fractals.

I used a Weierstrass level dimension in a Weierstrass to get a new kind of fractal in that scaled of of harmonic functions as well. By using a function that limited the dimension to a domain of [1/2,1] I was able to get past the problems at low dimension in Weierstrass functions:
 

21) s=Max(abs(s),1-abs(s))

So I had this kind of self-similar dimension working in two completely different kinds of fractals.

This new level of chaos isn't easy at the two dimensional level. In most cases involving just two independent random transforms one gets space filling blobs with very little defining pattern. To get a definite pattern in not one but three of the traditional  IFS fractals using three randomly switched independent transforms is a new level of complexity. I don't claim that this kind of information is related to the prime sequence; only that study of the prime sequence led me to the experiments to gave me this new chaos.

True basic Sierpinski new chaos:

PRINT "input Sierpinski number 2 to 12"

INPUT m

SET MODE "color"

SET WINDOW 0,1026,0,750

SET COLOR MIX (1) 0,0, 0

SET BACKGROUND COLOR "white"

LET x=1

LET y=1

LET c=0

LET s1=200

LET s2 =(s1)*750/1026

DIM a(25),b(25)

FOR i=1 to m

LET a(i)=cos(2*pi*i/m)

LET b(i)=sin(2*pi*i/m)

NEXT i

PRINT " SIERPINSKI I.F.S. New Chaos "

PRINT " SYMMETRICAL DIMENSIONAL SELF SIMILAR"

PRINT " BY R.L.BAGULA 2 Oct 2001(C) copy rights reserved"

RANDOMIZE

PRINT " M=";m

LET s=0.5

FOR n= 1 TO 2000000

LET c =RND

LET d=RND

IF rnd>=0.5 then LET s=abs(s/2-1/2) else LET s=s/2+1/2

LET s0=s

LET l=1+int(c*m)

LET k=1+int(d*m)

LET x1=sgn(x)*abs(x/2)^s0+sgn(a(l))*abs(a(l))^s0

LET y1=sgn(y)*abs(y/2)^s0+sgn(b(k))*abs(b(k))^s0

LET x=x1

LET y=y1

SET COLOR 255

IF n>10 THEN PLOT 1026/2+s1*x,750/2+s2*y

NEXT n

END

True basic Weierstrass new chaos function:

100 SET MODE "color"

110 SET WINDOW 0,1026,0,750

120 SET BACKGROUND COLOR "white"

130 SET COLOR MIX(1) 0,0,0

140 LET n=m*750/1026

150 PRINT" Weierstrass function"

160 PRINT " WITH WEIERSTRASS MAXIMAL DIMENSIONAL FUNCTION "

170 PRINT " by R. L. Bagula 2 OCT. 2001(C) copy rights reserved"

180 LET s0=log(2)/log(3)

190 FOR w=-pi/2 to pi/2 step 2*pi/(30000*2)

200 LET sx=0

210 LET sy=0

211 LET ss=0

220 FOR k=1 to 25

221 LET ss=ss+2^(-s0*k)*sin(w*(2^k))

222 LET sa=max(abs(ss),1-abs(ss))

230 LET sx=sx+2^(-sa*k)*cos(w*(2^k))

240 LET sy=sy+2^(-sa*k)*sin(w*(2^k))

248 NEXT k

280 LET x1=(1026/2)+ 1026*sx/2.5-50

290 LET y1=(750/2)+ 750*sy/2.5

300 SET COLOR 1

310 PLOT x1,y1;

320 NEXT w

330 PLOT x1,y1

340

350 END

True basic Prime information program:

100 SET MODE "color"

110 SET WINDOW 0,1024,0,750

111 SET COLOR MIX(0) 1,1,1

112 SET COLOR MIX(1) 0,0,0

120 REM sieve

121 PRINT "running sieve takes time"

130 LET E=640*4

140 DIM P(2500)

150 LET Q=0

160 FOR N=1 To 8*E

170 IF N<4 Then

172 LET Q=Q+1

174 LET P(Q)=N

178 GOTO 280

179 END IF

180 LET I=0

190 LET T=2

200 LET J=Int(N/T)

210 LET K=J*T

220 IF N=K Then GOTO 280

230 LET I=I+1

240 LET L=T*T

250 IF L>N Then

252 LET Q=Q+1

253 IF q>e then GOTO 300

254 LET P(Q)=N

258 GOTO 280

259 END IF

260 LET T=I*2+1

270 GOTO 200

280 IF Q>E Then GOTO 300

290 NEXT N

300 CLEAR

301 PRINT" Prime information: entropy and expectation"

302 Print" by Roger L. Bagula 3 Oct 2001(C)"

310 for x=1 to 1024

320 let sum=0

321 let sum1=0

330 for n=1 to x

340 let sum=sum+(-1)^n*log(p(n))/p(n)

350 if n>1 then let sum1=sum1+log(n)/log(P(n))

360 next n

370 if P(x)>1 then Plot sum1,750/2+750*sum;

371 let s=s+sum

372 let count=count+1

373 let ss=ss+sum1

374

380 next x

381 print sum,s/count,sum1/count, (s+sum1)/count

460 END

Last Modified:by Roger L. Bagula 3 Oct 2001©