Can a Differential Equation Describe the Prime Counting Function?

[eljose]

Can a differential equation for $$\pi (x)$$ (prime number counting function ) exist?..for example of the form

$$f(x)y'' +g(x)y' +h(x)y = u(x)$$ where the functions f,g,h and u

are known, and with the initial value problem $$y(2)= 0$$ for example...or is there any theorem forbidding it?..

By the way do you Number theoritis use Numerical methods ? (to solve diophantine equations, or Integral equations of first kind involving important functions) that,s all...

-In fact for every Green function of Any operator if we put:

$$\sum_ p L[G(x,p)] = \pi ' (x)$$ the problem is if some valuable info can be obtained from here

 

[shmoe]

pi(x) is a step function, it's derivative is zero everywhere except at primes where it is undefined.

 

[CRGreathouse]

You are perhaps thinking of Lagarias and Odlyzko?

J. C. Lagarias and A. M. Odlyzko, Computing pi(x): An analytic method, Journal of Algorithms, Vol. 8 (1987), pp. 173-191.

 

[saltydog]

How about this: Rather than looking at $\pi(x)$, how about considering what type of differential system would have as one or more of its solutions either the real or imaginary part of the zeta function on the critical line?

 

[imre mikoss]

Yes, I have found one

Dear eljose,
in a paper recently posted to a preprint server, I show how to find a Diff.
equation for Pi(x). Actually is a d.e. for the inverse of Pi(x). It is based in the fact that the sieving process produce symmetrical patterns between sieved a non-sieved numbers in N. The paper is non-technical because I am a physicist. I give no strict proofs. You can read the details in,
http://www.ma.utexas.edu/mp_arc/c/06/06-314.pdf

regards

Imre

 

[Playdo]

Dear eljose,
in a paper recently posted to a preprint server, I show how to find a Diff.
equation for Pi(x). Actually is a d.e. for the inverse of Pi(x). It is based in the fact that the sieving process produce symmetrical patterns between sieved a non-sieved numbers in N. The paper is non-technical because I am a physicist. I give no strict proofs. You can read the details in,
http://www.ma.utexas.edu/mp_arc/c/06/06-314.pdf

regards

Imre

I'll take a look at that. I'm not sure I am obliged to let a physicist skip out on rigor though.

On the original post, why a differential equation? If you are going continuous then I would think you want to consider the complex numbers as your domain and range. But, why would there not be a discrete analog in the realm of difference equations? The question is really asking if there are hidden variables behind the distribution of primes. I am inclined to think yes maybe, but that is based on very incomplete knowledge of some of the work of those how study ensembles of random matrices.