Pi(x) function in number theory solved¡¡¡

[eljose79]

I think i have solved the problem of getting the pi(x) function in number theory..i have tried to submit to several webpages but have been rejected so my last resort is to submit to this page hoping that someone give me an oportunity.

I am submiting this file from my universty...i have no unix..so the snob do not let me publish anything, i will know my ieas are corrects when i see a .pdf file edited by a famous mathematics about the smae topic..but remember the date i edit this message at 13-8-2003

In fact in the .doc paper the pi(x) function is discused and also its application to get all the primes hope you like it..i think is correct..i publish at physicsforums.org because i do not have another chance.

 

[suffian]

I don't know more than Calculus I and II, so you certainly don't have to take my opinion to heart; but, I really doubt your solution works as proposed (my limited knowledge doesn't allow me to check it).

What puts me off? Well, firstly, I can easily go on the net and find sources that state great mathematicians (including Gauss and Riemann) took a stab at making sense out of pi(x). These sources state that they did indeed notice patterns and methods of approximating the function quite well, but they didn't find any golden key that immediately states pi(x) given x. To say that you have solved this mystery in a single page of work using methods that (and I might be wrong on this) should be well known by many undergraduate math students sounds a little far-fetched.

Ofcourse, I don't think anyone should simply scoff at a piece of work because they doubt it will be solved by a such an such a person or in such an such a way. Your work deserves to be looked over. But, have you actually plugged in values for x (perhaps by using a computer program) to verify the values of pi(x) from your formula agree with table values found on the net and elsewhere (for large and small values)? Also, there are numerical methods for calculating pi(x) that are fast and accurate. Are you sure your formula isn't simply similar to one of these methods (note that these methods can find pi(x) for impressively large numbers)?

 

[plus]

I just had a quick look through the paper. However, I would prefer it if you would put limits on your summations (and variables to be summed), and similar for the integrations.

 

[eljose79]

I will try to improve it

i will try to improve the paper by editing it in .pdf format with an equation editor...by the way do you know another place to send it appart form arxiv.org?..

 

[plus]

Some preliminary questions:

1. Where does the identity you labelled (1) come from?
2. If x is not an integer, is Pi(x) still defined in the same way?
3. An approximation of Pi(x) is not the same as Pi(x) (alluded to in (3)). Why did you not leave the expression exact?

 

[Broken]

Convergence

Can it be proven the Laplace transform of Pi(x) even exists (i.e. converges)? If it can't be done rigorously nothing after (2) is valid. Furthermore, you really don't want to do an approximation if you want to find integers.

On top of that, did you use your method to do what it is supposed to do and does it work?

Broken

 

[Hurkyl]

IIRC, a function has to grow exponentially (or faster) in order for the Laplace transform not to exist. Pi(x) grows as ln(x), so its transform will exist.

 

[eljose79]

replies

1.The Pi(x) function diverges as x/lnx so when taking the limit
x tends to infinite of exp(-sx)x/Ln(x) it converges.

2.The method can not give and exact result...it only provides an aproximate result exact values can be calculated but the computation effort is very big

3.Pi(x) function for x not an integer has a meaning in that way:
Pi(x) when x not an integer should be equal to Pi([x]) where [x] means the integer part of x

A better version of the file is included in .pdf format made with an equation editor.