# Curious identity?

1. Mar 20, 2006

### eljose

Euler proved the identity:

$$\product_{p}(1-p^{-s})=1/\zeta(s)$$

he made use of the fact that f(x)=x^{-s} was multiplicative..but what would have happened if f(x) were of the form:

$$f(x+y)=f(x)*f(y)$$ (1)

then if Goldbach conjecture were correct we have that :

$$p1+p2=Even$$ $$p3+p4+p5=Odd$$ for n>5 then for our function f(x):

$$f(p1+p2)=f(2n)=f(p1)*f(p2)$$

$$f(p3+p4+p5)=f(2n´+1)=f(p3)*f(p4)*f(p5)$$s o i think we would

have that for every function satisfying (1) we should have that:

$$\product_{p}(1+f(p))=\sum{f(n)}+ R$$

where the sum is almost all positive integers but 0 and others that can,t be represented as a sum of primes these terms are included inside the R term.

Of ocurse this is not truly rigorous is only intuitive..:tongue2: :tongue2: :tongue2: i,m not completely sure if this all is true.

Last edited: Mar 20, 2006