## the prime number distribution in R

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.

 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.

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 on

Now 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.