The Function Pi(x) formula (aproximate)

In summary, the conversation discusses a method for obtaining the formula for Pi(x) by using the Laplace transform and an alternate series. The formula is submitted to universities but has not received a response. There may be some errors in the method or snobism may be a factor in not accepting the contribution.
  • #1
eljose79
1,518
1
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...
 
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  • #2
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).
 
  • #3


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.
 

1. What is the Pi(x) formula and what does it represent?

The Pi(x) formula is a mathematical function that calculates the number of prime numbers less than or equal to a given value of x. In simpler terms, it represents the number of prime numbers within a specific range.

2. How is the Pi(x) formula different from the traditional way of counting primes?

The traditional way of counting primes involves manually checking each number to see if it is divisible by any number other than itself and 1. This method becomes impractical for larger numbers. The Pi(x) formula provides a more efficient and accurate way of counting primes by using mathematical equations.

3. Can the Pi(x) formula be used to find the exact number of primes within a range?

No, the Pi(x) formula only provides an approximation of the number of primes within a given range. It becomes more accurate as x gets larger, but it will never give an exact count.

4. How is the Pi(x) formula useful in mathematics and science?

The Pi(x) formula is used in various fields of mathematics and science, such as number theory, cryptography, and computer science. It helps in identifying patterns in prime number distribution and can be used in algorithms for solving complex problems.

5. Are there any limitations or weaknesses of the Pi(x) formula?

Yes, the Pi(x) formula has its limitations, especially for smaller values of x. It may underestimate the number of primes, and it is not suitable for finding the exact number of primes. It also does not provide any information about the specific prime numbers within a given range.

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