Euler product and Goldbach conjecture

[eljose]

In an anlaogy with the Euler product of the Riemann function we make:

\prod_{p}(1+e^{-sp})=f(s) of course we have that:

f(p1+p2+p3)=f(p1)f(p2)f(p3) f(x)=exp(-ax) if Goldbach Conjecture is true then p1+p2= even and p5+p6+p8=Odd for integer n>5? then this product should be equal to:

f(s)=\sum_{n=0}^{\infty}a(n)e^{-sn} where the a(n) is the function that tells in how many ways and odd or even number can be descomposed as a sum of 2 or 3 primes, we now define:

A(x)=\sum_{n=0}^{x}a(n) if a(n)=1 for every n then using Perron formula we get that A(x)=[x] (floor function) so we would have that:

f(s)=(1-e^{-s}) then if correct take numerical values to proof that the product and f(s) are equal.

 

[eljose]

sorry there,s a mistake if we set:

\prod_{p} (1-e^{-sp}=f(s)= \sum_{n=0}^{\infty} a(n)e^{-sn}= \int_{0}^{\infty}A(x)e^{-sx}

And using the same trick Riemann did we have that:

f(s)= \sum_{n=1}^{\infty} \frac{g(x^{1/n})}{n}

where g(x)=\pi (lnx) so from this we could conclude that:

f(x)= \frac{1}{2 \pi i}\int_{C} ds \frac{ ln \zeta(s) }{s}ln^{s} (t)

Where C is the real line Re(s)>c for a real c constant