# Correspondence principle?

1. Jul 6, 2006

### eljose

"Correspondence" principle?..

let be the Zeta function defined for twin primes:

$$Z(s)=\prod (1-p^{-s})^{-1}$$ s>1

Where the product is taken only over twin primes so p and p+2 are primes..then my question is if we can define the Chebshev functions for twin primes $$\psi_2 (x)$$ $$\theta_2 (x)$$ in the form:

$$\theta ' _2 (x) = \pi '_2 (x)log(x)$$

$$\frac{Z'(s)}{Z(s)}=s\int_{0}^{\infty}dx\psi _2 (x) x^{-s-1}$$

the "correspondence2 principle in Number Theory would mean that we can define analogue functions for twin primes and normal primes....

Where Pi2(x) is the density of twin primes... if also is satisfied that:

$$\psi _2(x)- \theta _2 (x) =0$$ when x--->oo an asymptotic formula for twin primes density can be given.

Last edited: Jul 6, 2006