The prime number distribution in R

[ilario980]

Hi,
on my site http://ilario.mazzei.googlepages.com/home I've published a pdf containing the prime numbers distribution in R


Ilario Mazzei

 

[trambolin]

proof => noun
to prove => verb

and move this thread to... errm... say, number theory.

 

[ilario980]

i know that my english is not perfect, but I'm more interested in comments about the article - i published it in this forum to give more visibility to a revolutionary function in number theory (and not only)

 

[trambolin]

thank you ilario, you should also publish this for wider audience.

 

[ilario980]

thanks to you.Ilario M.

 

[willem2]

Is theorem 1 valid if a and b aren't integers?

what does If x -> $\infty$ then x is not a prime mean?

 

[ilario980]

Is theorem 1 valid if a and b aren't integers?

theorem 1 is valid in the domain of functions f and g, even if they are not integers (graphs in figures 5 and 6 has been drawn using this theorem)

what does If x -> $\infty$ then x is not a prime mean?

observe the graph of $$\Pi$$ function (figure 7); on each prime number greater than 0 the left limit is not zero. The limit of $$\Pi$$ function as x tend to +$\infty$ is zero, so +$\infty$ behaves like every not prime number.
In other words if x is infinitely great then x is a not prime number.

Ilario M.

 

[HallsofIvy]

You are going to have to be more precise on what you mean by "infinitely great". In all the extensions of numbers to include "infinitely great" numbers that I know, there are no "infinitely great" integers and so, of course, they are not prime. "Prime" is only defined for (finite) integers.

 

[ilario980]

using the $\Pi$ function (illustrated in figure 7 of my article) every prime number is a point of discontinuity where the left limit is different from 0, and every not prime number has the left limit equals to 0;
to understand why this happens please graph this products :

$$\prod sin(\frac{\pi * x}{j})$$

1) j from 1 to (5-1)
2) j from 2 to (5-1)see what happens at x=5

the $\Pi$ function is the extension of partial product from $$x - \lfloor x-1 \rfloor$$ and x-1; i.e.

$\Pi(4.9)$ => j from 1.9 to 3.9
$\Pi(4.95)$ => j from 1.95 to 3.95
$\Pi(5.1)$ => j from 1.1 to 4.1

setting x0 integer, if x approaches x0 from left we are very close to product 2, while if x approaches x0 from right we are very close to product 1

using the $\Pi$ function we can extend the concept of not prime numbers over R:
x is a not prime number if the left limit of $\Pi(x)$ is equal to zero (so infinity is a not prime number because the left limit is zero).

Ilario M.

 

[trambolin]

ilario take it easy.

Just send it to a journal if you really think it is revolutionary or something. I accept the fact that I will go into the history books as the ultimate idiot. But for me, nothing makes sense in your "article".

Use sinc function which makes more sense if you like, but this is not going anywhere. Dude, even Obama is in it.

 

[ilario980]

a technique to extend partial product does not make sense?

...and of course, the document is a draft;

moreover you are out of topic: go away if you want to lose your time

 

[trambolin]

it does not extend anything, where does this self confidence come from? Read some harmonic analysis. but ... nevermind. My bad. Viva la revolucion

 

[ilario980]

my confidence comes from the fundamental theorem of discrete geometric calculus (search "discrete calculus").in my document I have given a precise definition of the $\Pi$ function; there are infinitely many primes: can you find at least one prime number (or not prime number) in the domain $(1,+\infty)$ where my function goes wrong?

 

[ilario980]

armonic analysis is a very interesting topic.

From wikipedia:

"When Fourier submitted his paper in 1807, the committee (which included Lagrange, Laplace, Malus and Legendre, among others) concluded: ...the manner in which the author arrives at these equations is not exempt of difficulties and [...] his analysis to integrate them still leaves something to be desired on the score of generality and even rigour."

 

[CRGreathouse]

OK. So your paper defines a complicated function Pi-tilde of x and gives definitions for three sets: the "prime numbers", the "not prime numbers", and the "semiprime numbers"; these sets are disjoint and have the extended real line as their union.

But the definitions do not correspond to the usual definitions of those terms. Certainly 1 is not a semiprime (it's 0-almost-prime), and 4 is both nonprime and semiprime. So you've invented terms but not showed how they correspond to their usual counterparts.

Another issue: it's not clear from the definition if there's even an effective algorithm for computing the classification for a number in your scheme.

Pitilde(97.1)

Pitilde(97.0000000001)

and

Pitilde(97.0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001)

are each about 9000, but if you get close enough you'll see that the limit actually goes to 0. So how far do you have to go, in general?

 

[ilario980]

using the Pi-tilde function i have classified numbers in the following way:if the left limit of Pi-tilde(x) is zero then x is a not prime number;

if the left limit of Pi-tilde(x) is not zero then

if the right limit of Pi-tilde(x) is zero then x is a prime number
else
x is a semiprime number
​

on 0 and 1 the pi-tilde is continuous and the limit is 1(the pi-tilde function is not defined on integers)

i don't understand your second question; here is the graph of pi-tilde(x) near x=97:

http://ilario.mazzei.googlepages.com/Immagine.jpg Ilario M.

 

[willem2]

theorem 1 is valid in the domain of functions f and g, even if they are not integers (graphs in figures 5 and 6 has been drawn using this theorem)

Unfortunately I have no idea what to make of the infinite product, if a and b are not integers, so the meaning of theorem 1 isn't clear to me. I don't think there is any standard meaning to a summation or product with non-integer limits.

how should I compute

$$\prod_{x=1.3}^{2.8} x$$

for example?

 

[ilario980]

partials products are defined on integers (as partial sums). Theorem 1 says that if f e g are two functions such that:

$$\frac{g(x+1)}{g(x)}=f(x+1)$$


then the partial product from a to b of f(x) is:

$$\frac{g(b)}{g(a-1)}$$


to verify this theorem for f(x)=x then you have got to use the $\Gamma$ function (http://en.wikipedia.org/wiki/Gamma_function)

in particular:

$f(x)=x$

$g(x)= \Gamma (x+1)$

so using g(x) we can extend partial products in the domain of functions f and g


$$\prod_{x=1.3}^{2.8} x = \frac{g(2.8)}{g(0.3)}=\frac{\Gamma(3.8)}{\Gamma(1.3)}\approx 5.23$$


please let me know if you have any other question

Ilario M.

 

[hamster143]

Page 7, it's not clear what k is and where it comes from; there are two different k's (with and without subscript) and their meaning is never elucidated.

Page 9, when you say "function $$\Omega$$ as enunciated in the next section", presumably you mean previous section.

The relationship of the whole construct to the distribution of prime numbers is not clear. You have constructed a function that behaves in a certain way around primes, you don't draw any conclusions about prime number distribution, and it's not clear if any can be drawn. Even the statement from page 12 that the number of primes is "countable infinite" is not a consequence of any developments in the paper.

 

[ilario980]

in my paper i intentionally leaved some points without explanation; this to avoid that someone attibute himself the discovery of the pi-tilde function(if this has offended in some manner i apologize).

Page 7, it's not clear what k is and where it comes from; there are two different k's (with and without subscript) and their meaning is never elucidated.

suppose we want to know an approximation of partial products

by theorem 1 we can say that g(x+1)=g(x)*f(x), but of course, we don't know g(x)

what we can say is that if g(x) exists than must exist a constant k such that g(0)*k=1
in this way we can know every value of g(x)*k where x is a integer:

k*g(0)=1
k*g(1)=k*g(0) * f(1)

and so onNow let's evaluate g(0.1)
again we don't know the value of g(0.1) but, if g(0.1) exists than must exist a constant k(0.1) such that

k(0.1)*g(0.1)=1
k(0.1)*g(1.1)=k(0.1)*g(0.1) * f(1.1)this can be viewed in figure 5

Page 9, when you say "function LaTeX Code: \\Omega as enunciated in the next section", presumably you mean previous section.

in that section i have given the definition of pi-tilde function using theta function.
Partial products are taken with a distance that is integer (from 1.3 to 5.3, from 1.4 to 5.4)
in this way the k(x) constant vanish in the division
however, the goal of the document is to establish where prime numbers are distributed in R (i have not made conclusions about the magnitude)Ilario M.