Re: product convergence
- --- In firstname.lastname@example.org,
Kermit Rose <kermit@...> wrote:
> what is wrong with my "Proof"It began OK and then fizzled out.
Here is an elementary proof.
Proposition: The product prod(p>2, (p-2)/p) over primes p>2 vanishes.
Proof: It suffices to show that the contrary proposition is absurd.
Suppose that prod(p>2, (p-2)/p) did not vanish.
Then, by taking logs, we would conclude that
sum(p>2, log(p) - log(p-2)) is finite. But
log(p) - log(p-2) > 2/p. Thus
sum(p>2, 1/p) would also be finite.
Yet that is easily proven to be absurd:
Hence the product prod(p>2, (p-2)/p) does indeed vanish.
Comment: By the same argument, it follows that
the product prod(p>x, (p-x)/p) over primes p > x
vanishes for all real x > 0.