Integral representation of Pi(x)/x^4

[eljose79]

here your are my last contribution to number theory, i tried to send it to several journals but i had no luck and i was rejected, i think journals only want famous people works and don,t want to give an oportunity to anybody.

the work is attached to this message in .doc format only use Mellin transform method and complex integration hope you can find interesting.

 

[matt grime]

Jose, the reason that your 'paper' got rejected is because it is not new maths, it is not research to say that certain functions have a certain alternate representation that is simply a transform. Every l^2 function on T^1 has a Fourier series for instance, writing a Fourier series out is not research, especially as you don't even evaluate any of the damn integrals, ever. stop being insulted by journals rejecting you and try doing something original if you want to be published

 

[eljose79]

but if you give a integral that allows you calculate the prime number counting function..this would be new wouldn,t it?

 

[matt grime]

No, since yo can't actually do the integral. Or are you claiming to have done the integral and summed the infinite resulting series? No finite partial sum is acceptable unless you know the large scale behaviour. You've never discussed convergence issues or any of the properties of the terms in the series. I can give a partial difference equation, the solution to which will give the prime counting function. I can give several algorithms to calculate the prime counting function. This isn't new. You need to demonstrate that you've got something good. You've failed to do this, and never answered any single problem raised over the work.

 

[eljose79]

But the integral can be calculated using the residue theorme or by numerical methods...so i don,t think it can be useful,it,s only an integral

By the way could you tell me where co7uld i find the finite -diference equation for prime number counting function..thanks.

 

[eljose79]

Sorry sometimes i make mistakes in writting, i meant that you could obtain the integral by using numerical methods,in fact the other approximation is made by integratin from 2 to x the function t/ln(t) which can be only calculated numerically...i,m on the same case i think still the problem of "not-beiong-a-famous-mathematician" exists.

 

[Hurkyl]

You really should try learning about computational complexity and numerical approximation.

That you suggest integrating t / ln t is somehow comparable to your triple integral over the complex plane involving zeta functions demonstrates that you really don't understand the issues involved here.


One difference is that the former is simple enough that college freshmen are expected to be able to be able to figure out how much work is needed to get a good approximation. Yours is so complicated I don't even know where to begin trying to figure out the work involved.

 

[matt grime]

And you can't even figure out a difference equation whose solution gives the prime counting function?

Here's the easiest one:

x(n) = x(n-1) + f(n)

where f(n) =0 if n is composite and 1 otherwise, with boundary condition x_2 = 1.


This is of course completely useless, but it still gives the prime counting function in a far more elegant and exact way than anything you've written down.

 

[eljose79]

-Mine is exact whereas the integral t/ln(t) is only an approximation,the Pi(x) is not easy to obtain i am giving an exact formula to calculate it.

-yes Matt but you have x(n)-x(n-1)=f(n) you don,t know how is the form of f(n) whereas the functions involved in my integral are known functions (involving Riemann,s function)

 

[matt grime]

So, you know zeta(t) for all t? where do the zeroes lie?

If you're so certain of your 'solution', why not work out pi(10^10)?

I know exactly what f(n) is for all n. You don't know what zeta(pi) is do you? I mean you don't have an exact decimal expansion of its real (and complex) part.

I don't think you understand the difference between exact and approximate at all, do you?

So, the challenge is quite simpe. Find pi(10^10), indeed find pi(100). I can write a computer program using my very poor algorithm/solution that'll return the answer in a very short time (for 100). I can drastically improve it using Lagranges method, and improve that with Meissel's formula, and I can improve on all those too. Heck, even usenet troll James Harris has a program that will calculate pi(n) for reasonably small n in reasonable time.

 

[Hurkyl]

I think one of the hardest facts to really understand about numerical computation is that numeric operations simply are not analytical operations. A friend of mine has a great quote that succintly and simply describes this fact:

"(Numerical) addition is not associative!"

Once you come to terms with the fact that numerical computation doesn't even have addition right, it becomes easier to come to grips with the real difficulty of more complicated things.

 

[eljose79]

-Yes Matt,i am not good at computers but i am pretty sure that there is an algorithm to calculate my integral numerically (tell me if i am wrong and this integral can not be calculated numerically) so we wouldn,t need to know the zeroes of Riemann function.

In fact can you calculate f(n) for n=10^100000000000000000000000000000000

Another answer is that almost more than a billion of zeroes of Riemann Zeta function are known

 

[matt grime]

You once claimed to have an exact representation of the prime counting function, now you need to do a numerical integral. I have only algorithms, and am honest in saying that. So do your calculation, and stop claiming it is in anyway an exact answer, or indeed one that interests anyone if the journal rejections are anything to go by. Many relations are known for pi(x) already.


Perhaps you might care to read Tim Gowers's online essay on 'what is solved when one solves something'

f(n) = 0 for the number you cite since it is clearly composite. You do understand what f is don't you?

So what that more than a billion zeroes are known. Do you even know how many zeroes there are? (Currently it's something like 70% or so that are known to lie in the critical line, but this is neither here nor there.)

 

[matt grime]

So far I've been pointing out the basic practical limitations to your work, since there appears to be nothing theoretically amazing here in saying that some function posssesses some transform.

I'd like to go further in the theoretical vein now, and remind you that I have often asked you if you've ever checked if this has been done before. You've evidently not done that as, if you had done so, you'd know that Tate-Iwasara theory, itself over 50 years old, extensively uses the Fourier Analytic duality (and Mellin transforms) in its study of various L functions (and hence zeta functions), and that Wiener (I think I mean Wiener) published a book in 1941 where he uses Laplace transforms to answer some very basic and important questions in analytic number theory, particularly wrt to the prime number theorem.

So, apart from the practical issues and the fact that more than 50 years ago people were deriving much more complicated results using the same methods on the same objects, can you think of any other reason why it might have been rejected by the journal. Oh, yep, cos you're not famous enough...

 

[Hurkyl]

Watch your tongue.

I see no point in letting your gripe fest continue.