## Re: product convergence

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• ... 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
Message 1 of 8 , May 8, 2010
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:
http://primes.utm.edu/infinity.shtml#punchline
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.

David
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