The Function Pi(x) formula (aproximate)

[eljose79]

I think it is a good way to obtain the formula for Pi(x),where
Pi(x)={Sum over primes<x}of 1

We know that S(p)Exp(-sp)=Int(0,infinite)Pi(x)Exp(-sx)..where s is a parameter s>0 and S(p) means sum over all primes.

But Int(0,infinite)Pi(x)Exp(-sx) is just the Laplace transform (will be donoted by L) of the function Pi(x) so we would have.


S(p)Exp(-sp)=L(Pi(x)) or taking the inverse

L**(-1)Exp(-sp)=Pi(x). (1)

To get the sum S(p)Exp(-sp) we will use the formula

S(0,infinite){Pi(x)-Pi(x-1)+1}(-1)**nZ**n=2Z**2-S(p)Z**p+1/1+Z**p (2)

as you can check a(n)={Pi(n)-Pi(n-1)+1} we would have an alternate series so we could use an Euler transform to improve the convergence and from (2) we could obtain S(p)Exp(-sp) and substitute it into the formula given in (1) to obtain an approach to formula Pi(x).


I submitted it by e-mail and letter to several universities in my country (spain) but i did not have any answer..i do not know if there is something wrong in my method or..if it is useles...or if perhaps will be only snobism..to accept that a non-mathematician person this important formula...

 

[eljose79]

Oops..i made a mistake...

S(p)Exp(-sp)=sInt(0,infinite)Pi(x)Ex(-sx), this would be the correct formula..where Pi(-1)=Pi(0)=0=Pi(1).

 

[tacman]

Thank you for sharing your thoughts and approach to obtaining the function Pi(x) formula. Your method seems to be a valid approach and I can understand why you would like to share it with universities in your country. However, it is important to keep in mind that the formula for Pi(x) has been extensively studied and researched by mathematicians for centuries, and there may already be established and accepted methods for obtaining it. It is possible that your approach may have already been explored and may not be considered as groundbreaking as you may believe. It is also possible that universities may not have responded to your submission due to a variety of reasons, such as a lack of resources or a focus on other areas of research. I would suggest continuing to share your ideas and research with others, as it can lead to valuable discussions and collaborations. However, it is also important to acknowledge and respect the work of others in the field of mathematics.