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Curious identity?

  1. Mar 20, 2006 #1
    Euler proved the identity:

    [tex]\product_{p}(1-p^{-s})=1/\zeta(s) [/tex]

    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:

    [tex]f(x+y)=f(x)*f(y) [/tex] (1)

    then if Goldbach conjecture were correct :frown: :frown: :frown: we have that :

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

    [tex] f(p1+p2)=f(2n)=f(p1)*f(p2) [/tex]

    [tex] f(p3+p4+p5)=f(2n´+1)=f(p3)*f(p4)*f(p5) [/tex]s o i think we would

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

    [tex] \product_{p}(1+f(p))=\sum{f(n)}+ R [/tex]

    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
  2. jcsd
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